{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0001","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3d4d49526d338119faa041e5aabae79b1f584e88915837d8f6302eb9e02f3d7","family_id":"qf-mod33-kernel-membership","seed":202611022001,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,1],[2,2]],\"vector\":[-1,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 1], [2, 2]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-1, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-1, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-1, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0002","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"486e0adeda1736aeda5bb3f0f5de4bbbeca9dbdfa5611ee0a261c42b657bc96b","family_id":"qf-mod33-kernel-membership","seed":202611022002,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,2],[2,4]],\"vector\":[-2,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 2], [2, 4]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-2, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-2, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-2, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0003","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78c58fd7bdb9ab5021e12f10ceb9bde250d0bf87070315d7f9baeb93b4608871","family_id":"qf-mod33-kernel-membership","seed":202611022003,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,3],[2,6]],\"vector\":[-3,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 3], [2, 6]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-3, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-3, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-3, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0004","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3e609e81fc7fa5a3957d3da65b11bfbe0994d3f3cd4c445f444f08cc41a3ed1","family_id":"qf-mod33-kernel-membership","seed":202611022004,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,4],[2,8]],\"vector\":[-4,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 4], [2, 8]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-4, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-4, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-4, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0005","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6211f138a57b96769f76742ee65e835e7ce6f6196be905b1fc0abb63f490fe40","family_id":"qf-mod33-kernel-membership","seed":202611022005,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,5],[2,10]],\"vector\":[-5,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 5], [2, 10]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-5, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-5, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-5, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0006","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d925f6daecb28152a0251693e428b8e52bc25336e32415f60191331315c58ebb","family_id":"qf-mod33-kernel-membership","seed":202611022006,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,6],[2,12]],\"vector\":[-6,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 6], [2, 12]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-6, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-6, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-6, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0007","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c48d6684a4ab36f86c14d90e79dfbe9c198ff565f086d6690a038252742b043","family_id":"qf-mod33-kernel-membership","seed":202611022007,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,7],[2,14]],\"vector\":[-7,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 7], [2, 14]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-7, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-7, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-7, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0008","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92fd07ea38df71cf3d6d420753e3528df03ff19c2bcb60fa0f643e70d277aab9","family_id":"qf-mod33-kernel-membership","seed":202611022008,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,8],[2,16]],\"vector\":[-8,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 8], [2, 16]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-8, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-8, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-8, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0009","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9625ac4976a0586f5fbbc741927ea308afa08dcabaecceabc0a239c79540966c","family_id":"qf-mod33-kernel-membership","seed":202611022009,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,9],[2,18]],\"vector\":[-9,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 9], [2, 18]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-9, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-9, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-9, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0010","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a10eb35524e85d0739af6dca616382b146563bcf897ea92b49fe72d00218b383","family_id":"qf-mod33-kernel-membership","seed":202611022010,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,10],[2,20]],\"vector\":[-10,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 10], [2, 20]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-10, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-10, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-10, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0011","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bae87220f7231a9b8aeba0de4f37040d6d50f97b2100f5b8590adee49fbf53b1","family_id":"qf-mod33-kernel-membership","seed":202611022011,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,11],[2,22]],\"vector\":[-11,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 11], [2, 22]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-11, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-11, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-11, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0012","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0eb99913a3cb1de7c795912b7a4c7a7ac3d267f041b016750eb21ee9b13fda48","family_id":"qf-mod33-kernel-membership","seed":202611022012,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,12],[2,24]],\"vector\":[-12,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 12], [2, 24]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-12, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-12, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-12, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0013","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4fb9ef085367d6fcae06e4e123b2e98a2fb71a8d019ddb2f5438159caa4e68f","family_id":"qf-mod33-kernel-membership","seed":202611022013,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,13],[2,26]],\"vector\":[-13,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 13], [2, 26]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-13, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-13, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-13, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0014","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c23534e74f01442a2394207e3686a13b214a1902125d4bd754244487becaf60","family_id":"qf-mod33-kernel-membership","seed":202611022014,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,14],[2,28]],\"vector\":[-14,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 14], [2, 28]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-14, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-14, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-14, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0015","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67a826445346039276454958dd8666db07282460bb118e688a37f10ae01b2564","family_id":"qf-mod33-kernel-membership","seed":202611022015,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,15],[2,30]],\"vector\":[-15,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 15], [2, 30]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-15, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-15, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-15, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0016","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d240bab4081fef4f9919ea75310db52a9b638ca1ec2966b62c072ca272cc7fc","family_id":"qf-mod33-kernel-membership","seed":202611022016,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,16],[2,32]],\"vector\":[-16,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 16], [2, 32]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-16, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-16, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-16, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0017","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38547dc7d8eff63199b4fca20e74dbf9403bde3b7444f5126a9b9f9c392a8f0a","family_id":"qf-mod33-kernel-membership","seed":202611022017,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,17],[2,34]],\"vector\":[-17,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 17], [2, 34]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-17, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-17, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-17, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0018","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b907e3b6f902b5e9d674c075b724341ed15bc5eeef11172478a9d4565b59142","family_id":"qf-mod33-kernel-membership","seed":202611022018,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,18],[2,36]],\"vector\":[-18,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 18], [2, 36]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-18, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-18, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-18, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0019","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"039b7487c657aa80ff9ddbc071b248fe5e6e943e229a85984c888b9e0c6f4f21","family_id":"qf-mod33-kernel-membership","seed":202611022019,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,19],[2,38]],\"vector\":[-19,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 19], [2, 38]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-19, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-19, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-19, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0020","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7c0c8be673051520d801c2c274627c4fbe25cd6d8fcb65a858e83f1fa72b3c4","family_id":"qf-mod33-kernel-membership","seed":202611022020,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,20],[2,40]],\"vector\":[-20,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 20], [2, 40]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-20, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-20, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-20, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0021","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0b77207d12051cdf8d2742616e55a31c144c066c81b884ef4bece772c146fa6","family_id":"qf-mod33-kernel-membership","seed":202611022021,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,21],[2,42]],\"vector\":[-21,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 21], [2, 42]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-21, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-21, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-21, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0022","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9b9e1375cd0efab58ea9f7f15ef36710f70a444f2a7aca598f2b5ca40867eb2","family_id":"qf-mod33-kernel-membership","seed":202611022022,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,22],[2,44]],\"vector\":[-22,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 22], [2, 44]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-22, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-22, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-22, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0023","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0ad67c98bcdd6f3f667bd23b285d6e9a4e7f03ec5fb316a03930a8d25b8f44e","family_id":"qf-mod33-kernel-membership","seed":202611022023,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,23],[2,46]],\"vector\":[-23,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 23], [2, 46]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-23, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-23, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-23, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0024","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d37784b0cacc479727e79618c9755519ebede20fcd4aa933516fed4b8311882","family_id":"qf-mod33-kernel-membership","seed":202611022024,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,24],[2,48]],\"vector\":[-24,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 24], [2, 48]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-24, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-24, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-24, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0025","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95609a3a77c3997244761140810e7d38424957c9333f9b4ef7d85e6099784693","family_id":"qf-mod33-kernel-membership","seed":202611022025,"semantic_template":"qf-mod33-kernel-membership/{\"matrix\":[[1,25],[2,50]],\"vector\":[-25,1]}"},"fields":{"title":"Kernel Membership Has the Zero Vector as Its Target","stem":"Use this same exact integer matrix A=[[1, 25], [2, 50]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Selected vector=[-25, 1]. Give (vector,image,in_kernel?,kernel_target_is_zero_vector).","answer":{"kind":"expression","value":"([-25, 1], [0, 0], Yes, kernel_target_is_zero_vector)","explanation":"Compute A*x exactly and compare every component with zero. A vector mapping to a nonzero target belongs to that target fiber, not to the kernel. The exact result, retaining the requested order, is ([-25, 1], [0, 0], Yes, kernel_target_is_zero_vector)."},"skills":["Kernel Membership Has the Zero Vector as Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0026","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19dc0947792cb6ee4a1f3ce70191045b48380566da777da6183b35f095f162c0","family_id":"qf-mod33-kernel-basis","seed":202611022026,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,1],[2,2]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 1], [2, 2]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 1], [2, 2]], 1, [[1, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 1], [2, 2]], 1, [[1, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0027","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e53e15ab8306af7cda34f282d555160a68ad833648b69f6a8370554360a6424","family_id":"qf-mod33-kernel-basis","seed":202611022027,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,2],[2,4]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 2], [2, 4]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 2], [2, 4]], 1, [[2, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 2], [2, 4]], 1, [[2, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0028","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"378699d09e03a5de7ef57fe9485bd9497690aa9924b175c9dfa5a58d69bfe061","family_id":"qf-mod33-kernel-basis","seed":202611022028,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,3],[2,6]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 3], [2, 6]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 3], [2, 6]], 1, [[3, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 3], [2, 6]], 1, [[3, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0029","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfc0c38bfec1e1d4538020849ce908830189302919a6340b907821186b8e014c","family_id":"qf-mod33-kernel-basis","seed":202611022029,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,4],[2,8]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 4], [2, 8]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 4], [2, 8]], 1, [[4, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 4], [2, 8]], 1, [[4, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0030","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ec9ce8a689b2e48e1f405c3b1fc10e993ccc6ebaba7f14bbce470d50d968f91","family_id":"qf-mod33-kernel-basis","seed":202611022030,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,5],[2,10]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 5], [2, 10]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 5], [2, 10]], 1, [[5, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 5], [2, 10]], 1, [[5, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0031","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53c3fe7e0af587e00f33a93af5dbffcc006d52cc006ac06e39deaea3a2528072","family_id":"qf-mod33-kernel-basis","seed":202611022031,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,6],[2,12]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 6], [2, 12]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 6], [2, 12]], 1, [[6, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 6], [2, 12]], 1, [[6, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0032","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7acfa0042b2e5804b45d497818ed6ac029163cf2f49f4dcc0569e3f2b132fcd","family_id":"qf-mod33-kernel-basis","seed":202611022032,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,7],[2,14]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 7], [2, 14]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 7], [2, 14]], 1, [[7, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 7], [2, 14]], 1, [[7, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0033","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0605220b695ae7337ac7e4a07896460d2413e6b8f8d15c88c99eb0dfbac330d8","family_id":"qf-mod33-kernel-basis","seed":202611022033,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,8],[2,16]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 8], [2, 16]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 8], [2, 16]], 1, [[8, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 8], [2, 16]], 1, [[8, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0034","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34bb462abfc454358ce4d07a3f0bcc208538da24572569cbb526e05027b6edb9","family_id":"qf-mod33-kernel-basis","seed":202611022034,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,9],[2,18]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 9], [2, 18]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 9], [2, 18]], 1, [[9, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 9], [2, 18]], 1, [[9, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0035","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"852b0e5b35ee4612123516f45a04c3c6959628ee8bb0c77d6d146085e1366cf0","family_id":"qf-mod33-kernel-basis","seed":202611022035,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,10],[2,20]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 10], [2, 20]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 10], [2, 20]], 1, [[10, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 10], [2, 20]], 1, [[10, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0036","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"039a6999109c607219da9e90adf9bf372f1e4e12f100e85db0e57e9019187d40","family_id":"qf-mod33-kernel-basis","seed":202611022036,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,11],[2,22]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 11], [2, 22]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 11], [2, 22]], 1, [[11, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 11], [2, 22]], 1, [[11, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0037","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ad2b9c86bdc3c3db031a96b60398d952daf689e85825bd3edade6360d3e1526","family_id":"qf-mod33-kernel-basis","seed":202611022037,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,12],[2,24]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 12], [2, 24]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 12], [2, 24]], 1, [[12, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 12], [2, 24]], 1, [[12, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0038","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5db1757ae2c083edd68c4a26854e5bdd5153ceca234748bb3286c1044f2c5915","family_id":"qf-mod33-kernel-basis","seed":202611022038,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,13],[2,26]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 13], [2, 26]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 13], [2, 26]], 1, [[13, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 13], [2, 26]], 1, [[13, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0039","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78c5efcc648642a7c21b63f8783eb78ca28440ff204046da2def8bc56ebae390","family_id":"qf-mod33-kernel-basis","seed":202611022039,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,14],[2,28]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 14], [2, 28]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 14], [2, 28]], 1, [[14, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 14], [2, 28]], 1, [[14, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0040","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bf75c54dabd3db4551d1a016852203a47f8d465dbf0ee7e8b9625806490e040","family_id":"qf-mod33-kernel-basis","seed":202611022040,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,15],[2,30]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 15], [2, 30]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 15], [2, 30]], 1, [[15, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 15], [2, 30]], 1, [[15, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0041","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83f8a37227013b59d2f20deb47cec13b09e4e04fd9f8a295f7405e71f93a757c","family_id":"qf-mod33-kernel-basis","seed":202611022041,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,16],[2,32]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 16], [2, 32]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 16], [2, 32]], 1, [[16, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 16], [2, 32]], 1, [[16, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0042","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4382b2c8cf1ac293892f5cc5d11b41de543db20280d64e6d4adfeb0c30f7e276","family_id":"qf-mod33-kernel-basis","seed":202611022042,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,17],[2,34]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 17], [2, 34]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 17], [2, 34]], 1, [[17, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 17], [2, 34]], 1, [[17, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0043","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"483a453e9605a8ad28132387b8e868970f937b536885ae73b895d902b30b1ac7","family_id":"qf-mod33-kernel-basis","seed":202611022043,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,18],[2,36]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 18], [2, 36]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 18], [2, 36]], 1, [[18, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 18], [2, 36]], 1, [[18, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0044","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38ab408777a38c34b87393dd6c6d49a60ccba772024057a574fe9b9c16e960d3","family_id":"qf-mod33-kernel-basis","seed":202611022044,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,19],[2,38]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 19], [2, 38]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 19], [2, 38]], 1, [[19, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 19], [2, 38]], 1, [[19, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0045","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5b739f9fc6444823ff999a64a8bceeedf6869b2ebb6104b8658d0a5e5ee6195","family_id":"qf-mod33-kernel-basis","seed":202611022045,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,20],[2,40]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 20], [2, 40]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 20], [2, 40]], 1, [[20, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 20], [2, 40]], 1, [[20, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0046","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec04cda42fbafbab76dbad43a417118cda7c83281abb2d75e33a6fff7896007a","family_id":"qf-mod33-kernel-basis","seed":202611022046,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,21],[2,42]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 21], [2, 42]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 21], [2, 42]], 1, [[21, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 21], [2, 42]], 1, [[21, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0047","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17bf4f7f769f9caf6fa5b3eba7603cbd4de01301510ea0c7c719ef23b898819e","family_id":"qf-mod33-kernel-basis","seed":202611022047,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,22],[2,44]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 22], [2, 44]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 22], [2, 44]], 1, [[22, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 22], [2, 44]], 1, [[22, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0048","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca180180dfa3acb854fea807e93d72eb24f2688e8eeca7190eecf0b91cea9632","family_id":"qf-mod33-kernel-basis","seed":202611022048,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,23],[2,46]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 23], [2, 46]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 23], [2, 46]], 1, [[23, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 23], [2, 46]], 1, [[23, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0049","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26efae729d63b976ee50ea4bd63a19bd1550e3a3d059abe6f9c9bc6d6ffc139f","family_id":"qf-mod33-kernel-basis","seed":202611022049,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,24],[2,48]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 24], [2, 48]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 24], [2, 48]], 1, [[24, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 24], [2, 48]], 1, [[24, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0050","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e17bce38216f702f534d8e82b8f23bc004c30fcc59a2f9f7dcc2c32fcda5fc6e","family_id":"qf-mod33-kernel-basis","seed":202611022050,"semantic_template":"qf-mod33-kernel-basis/{\"matrix\":[[1,25],[2,50]]}"},"fields":{"title":"Give a Basis of the Entire Real Kernel, Not a Finite Search Result","stem":"Use this same exact integer matrix A=[[1, 25], [2, 50]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (matrix,rank,kernel_basis,nullity,symbolic_full_real_kernel).","answer":{"kind":"expression","value":"([[1, 25], [2, 50]], 1, [[25, -1]], 1, symbolic_full_real_kernel)","explanation":"Solve the homogeneous equations over the reals using the rank cases and the stated canonical basis convention. The displayed basis represents all real linear combinations, not merely its integer multiples. The exact result, retaining the requested order, is ([[1, 25], [2, 50]], 1, [[25, -1]], 1, symbolic_full_real_kernel)."},"skills":["Give a Basis of the Entire Real Kernel, Not a Finite Search Result"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0051","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d4b755a68339cce283f9d6e8fcb90514b6fe705adf2c85f711c15098dd5441e","family_id":"qf-mod33-kernel-nullity","seed":202611022051,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,1],[2,2]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 1], [2, 2]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[1, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[1, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0052","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"140321f06b835ab85e0ce20382d695dd002ed5de10ccf5f92d68bbaa8bfd7c91","family_id":"qf-mod33-kernel-nullity","seed":202611022052,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,2],[2,4]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 2], [2, 4]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[2, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[2, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0053","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72a6fb533e5afef144ef6060267c1b3871830fb6521d452dc54bfedaf17c72d7","family_id":"qf-mod33-kernel-nullity","seed":202611022053,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,3],[2,6]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 3], [2, 6]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[3, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[3, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0054","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68f8a79d56f85f8327fecd7b2f28ae5b79f509bf3d6da0d4ab730d63068dda6b","family_id":"qf-mod33-kernel-nullity","seed":202611022054,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,4],[2,8]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 4], [2, 8]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[4, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[4, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0055","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac46789c21bff6389c1a13525a78f90d5f4613909c6d0bf403d8f2b4ebb1af5b","family_id":"qf-mod33-kernel-nullity","seed":202611022055,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,5],[2,10]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 5], [2, 10]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[5, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[5, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0056","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdda0ea62f24ec47e34bca37f33d6d1c44c82ccb194d2d2ea366ad9e7617b230","family_id":"qf-mod33-kernel-nullity","seed":202611022056,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,6],[2,12]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 6], [2, 12]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[6, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[6, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0057","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"525164a251be4db8fc04b0c15572d3774b5cc3c9b1d37b9e6a7feed43b2da1cc","family_id":"qf-mod33-kernel-nullity","seed":202611022057,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,7],[2,14]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 7], [2, 14]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[7, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[7, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0058","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c98bc038b2c6967ee354a3627c40309da1c932b76cde903ef2ef27b12b2752d","family_id":"qf-mod33-kernel-nullity","seed":202611022058,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,8],[2,16]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 8], [2, 16]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[8, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[8, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0059","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7109ab89cf2a814e4ba7069c4a4070cb6eee41dd75a5e158bf96b25141080c0","family_id":"qf-mod33-kernel-nullity","seed":202611022059,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,9],[2,18]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 9], [2, 18]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[9, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[9, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0060","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74b374ef61d5886aed76b2032322e7a37ed1875b9dff1c18f0bbc974a72e0261","family_id":"qf-mod33-kernel-nullity","seed":202611022060,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,10],[2,20]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 10], [2, 20]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[10, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[10, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0061","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a627ede839a30fbba5dcfad051a2bd7180e9b40fe5f10fde8eee4ded4f083911","family_id":"qf-mod33-kernel-nullity","seed":202611022061,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,11],[2,22]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 11], [2, 22]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[11, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[11, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0062","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b1ac4daea4bc8873e2faeb53416761448ea4817f5bde99f35aa6a846aba86d6","family_id":"qf-mod33-kernel-nullity","seed":202611022062,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,12],[2,24]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 12], [2, 24]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[12, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[12, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0063","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0116aa90bdae40feaf456b5465d07a030a6bb58691d5b8adbef3e9a2b68b92c1","family_id":"qf-mod33-kernel-nullity","seed":202611022063,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,13],[2,26]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 13], [2, 26]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[13, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[13, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0064","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bec5695b14cc499822e5442522501d44d224e14399c7e857b7c550010e35de26","family_id":"qf-mod33-kernel-nullity","seed":202611022064,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,14],[2,28]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 14], [2, 28]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[14, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[14, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0065","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"820dc2b9fc119e1a543e4ce95000b8cc9f78690aef29ed7d479e0537e7a41db5","family_id":"qf-mod33-kernel-nullity","seed":202611022065,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,15],[2,30]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 15], [2, 30]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[15, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[15, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0066","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20bfad6f29298db5a3d3919221cb67de6a6b9625e8d0ef6d8efdda4a1eb69a49","family_id":"qf-mod33-kernel-nullity","seed":202611022066,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,16],[2,32]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 16], [2, 32]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[16, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[16, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0067","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0a5dcae49fa659c6cfdefcd8b6f4a34bb56aa94eda173232ca67504b559d73a","family_id":"qf-mod33-kernel-nullity","seed":202611022067,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,17],[2,34]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 17], [2, 34]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[17, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[17, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0068","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c67e64bca15fcb3162f171ef5ba9c416f799a9bdd77022e7824d6ca064d1a2ec","family_id":"qf-mod33-kernel-nullity","seed":202611022068,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,18],[2,36]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 18], [2, 36]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[18, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[18, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0069","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a021158359c569c4ac989f9a39716675d94552da07343ef60c2e5819fbc79019","family_id":"qf-mod33-kernel-nullity","seed":202611022069,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,19],[2,38]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 19], [2, 38]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[19, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[19, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0070","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a7ece2bbaca94b05ac472fb01cf5bc14eb8ae0b43be1f601d83d51aaa4f1aed","family_id":"qf-mod33-kernel-nullity","seed":202611022070,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,20],[2,40]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 20], [2, 40]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[20, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[20, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0071","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0816f4b3f8e528052a0a8edffecb81ed9c8010f50db864e50a99be409d8bfe7b","family_id":"qf-mod33-kernel-nullity","seed":202611022071,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,21],[2,42]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 21], [2, 42]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[21, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[21, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0072","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b45b36d795435e1e27aa507fc69257b4e2e21eb37eba36d981aaa4fd79309e0a","family_id":"qf-mod33-kernel-nullity","seed":202611022072,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,22],[2,44]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 22], [2, 44]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[22, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[22, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0073","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8621bb83a94bc46f0ca725ba27ff6d2d9df68a7d7351db2c58918e170e79d3ae","family_id":"qf-mod33-kernel-nullity","seed":202611022073,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,23],[2,46]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 23], [2, 46]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[23, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[23, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0074","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"315b2d03972497ff9b1544aedfabdfdaad6becec927c793741704c00e0937101","family_id":"qf-mod33-kernel-nullity","seed":202611022074,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,24],[2,48]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 24], [2, 48]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[24, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[24, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0075","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cbbca5587c70147723edb5da02aa6a561fe73ea7650dcc185133d106a22ea92","family_id":"qf-mod33-kernel-nullity","seed":202611022075,"semantic_template":"qf-mod33-kernel-nullity/{\"matrix\":[[1,25],[2,50]]}"},"fields":{"title":"Nullity Counts a Full Kernel Basis","stem":"Use this same exact integer matrix A=[[1, 25], [2, 50]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Give (rank,nullity,kernel_basis).","answer":{"kind":"expression","value":"(1, 1, [[25, -1]])","explanation":"Use nullity=2−rank for this two-dimensional source and retain the full symbolic real-kernel basis. The number of sampled grid points is not nullity. The exact result, retaining the requested order, is (1, 1, [[25, -1]])."},"skills":["Nullity Counts a Full Kernel Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0076","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8267a67fc972a1b10e9d7e1d0cb4da5ba2a1745a90679779fc3bb30978d114a2","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022076,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"matrix\":[[1,1],[2,2]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 1], [2, 2]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=1. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(1, [[-1, 1], [0, 0], [1, -1]], [[1, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (1, [[-1, 1], [0, 0], [1, -1]], [[1, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0077","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70758ad45593890be070a4518579769be3da34f59959eb494bbaefe32563890e","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022077,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":2,\"matrix\":[[1,2],[2,4]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 2], [2, 4]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=2. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(2, [[-2, 1], [0, 0], [2, -1]], [[2, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (2, [[-2, 1], [0, 0], [2, -1]], [[2, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0078","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f8990a273bd211ee57f9a086b75a3dd58005f780ae900eb975c97f555dcdea8","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022078,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":3,\"matrix\":[[1,3],[2,6]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 3], [2, 6]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=3. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(3, [[-3, 1], [0, 0], [3, -1]], [[3, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (3, [[-3, 1], [0, 0], [3, -1]], [[3, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0079","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb45cc5fd242d263bcf0ff94bfcc13919d1bf47eb99d7366968ab1fca39f6516","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022079,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":4,\"matrix\":[[1,4],[2,8]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 4], [2, 8]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=4. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(4, [[-4, 1], [0, 0], [4, -1]], [[4, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (4, [[-4, 1], [0, 0], [4, -1]], [[4, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0080","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ce3fba94480d0a14ad031099c46b0f4f18dd8063ce543689a62d955023f68c8","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022080,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":5,\"matrix\":[[1,5],[2,10]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 5], [2, 10]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=5. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(5, [[-5, 1], [0, 0], [5, -1]], [[5, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (5, [[-5, 1], [0, 0], [5, -1]], [[5, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0081","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"299b6a3a8578d390b8fa702cfcb9ad46a5dcceedd1fb2c662221b394e2a5fd68","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022081,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"matrix\":[[1,6],[2,12]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 6], [2, 12]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=1. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(1, [[0, 0]], [[6, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (1, [[0, 0]], [[6, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0082","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e43cd6c30f4b9e8961032edec084c725fc8cac9116c512a6a40536ea1bd5132","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022082,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":2,\"matrix\":[[1,7],[2,14]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 7], [2, 14]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=2. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(2, [[0, 0]], [[7, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (2, [[0, 0]], [[7, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0083","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a68dec3af63be35c63c72f3e22272f6b702e094c47c6e367e8b9e6edf835f18a","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022083,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":3,\"matrix\":[[1,8],[2,16]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 8], [2, 16]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=3. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(3, [[0, 0]], [[8, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (3, [[0, 0]], [[8, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0084","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67cb690f7a95b9d0c962dcc1d7b3113478923df389363324f669229b9a90e883","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022084,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":4,\"matrix\":[[1,9],[2,18]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 9], [2, 18]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=4. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(4, [[0, 0]], [[9, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (4, [[0, 0]], [[9, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0085","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d57d42a27d21d56f14f6654c788f7d0dc460fc6f729d6c99bd14acbab9e8c234","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022085,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":5,\"matrix\":[[1,10],[2,20]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 10], [2, 20]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=5. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(5, [[0, 0]], [[10, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (5, [[0, 0]], [[10, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0086","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63690b4802ff0bf9096a8e500a49330e11e300da958f6cf40ce262d4de7a0641","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022086,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"matrix\":[[1,11],[2,22]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 11], [2, 22]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=1. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(1, [[0, 0]], [[11, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (1, [[0, 0]], [[11, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0087","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20c8c72bc349a6504e63b03110ce195ead6d696e9232891ae349699e4493055d","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022087,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":2,\"matrix\":[[1,12],[2,24]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 12], [2, 24]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=2. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(2, [[0, 0]], [[12, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (2, [[0, 0]], [[12, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0088","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35444466744631a742e3e7d0d419fcf60eeff6346a442f556d7abf9aba910f13","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022088,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":3,\"matrix\":[[1,13],[2,26]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 13], [2, 26]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=3. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(3, [[0, 0]], [[13, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (3, [[0, 0]], [[13, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0089","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f51387b83e2edfe86d4a497bd1cf36f52ee1f6c400dc9847347f977089bd6a1","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022089,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":4,\"matrix\":[[1,14],[2,28]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 14], [2, 28]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=4. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(4, [[0, 0]], [[14, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (4, [[0, 0]], [[14, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0090","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66b2057a55b38c8ce406c41b7260c76e1da29a5e4549ada1d728e13f8a59632c","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022090,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":5,\"matrix\":[[1,15],[2,30]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 15], [2, 30]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=5. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(5, [[0, 0]], [[15, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (5, [[0, 0]], [[15, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0091","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cfc1195c693474839065e913d5dc29afc3d023bbabef0e6d992fc82d9e2191f","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022091,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"matrix\":[[1,16],[2,32]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 16], [2, 32]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=1. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(1, [[0, 0]], [[16, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (1, [[0, 0]], [[16, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0092","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eae85ede169570a567c1e131429c2b1586e8d08543ddd25c248474a6bb7e6844","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022092,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":2,\"matrix\":[[1,17],[2,34]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 17], [2, 34]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=2. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(2, [[0, 0]], [[17, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (2, [[0, 0]], [[17, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0093","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"beee252225bd12fb9f3e9249dee5191f1bb8b8477dd670a3095582347ffead2d","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022093,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":3,\"matrix\":[[1,18],[2,36]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 18], [2, 36]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=3. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(3, [[0, 0]], [[18, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (3, [[0, 0]], [[18, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0094","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c06148888f42322e197c383ca806fe8fd9df8ab76d8faf4ee3ca47134ea52583","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022094,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":4,\"matrix\":[[1,19],[2,38]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 19], [2, 38]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=4. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(4, [[0, 0]], [[19, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (4, [[0, 0]], [[19, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0095","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fef13244323ba7b756737ab7a1cb7e33911d67729287a39ada695bfd95bbcf8","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022095,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":5,\"matrix\":[[1,20],[2,40]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 20], [2, 40]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=5. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(5, [[0, 0]], [[20, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (5, [[0, 0]], [[20, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0096","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"820c183ef34698f28fd5f066ae9794e0b24f9ed99ec40d520408782f52dfadd2","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022096,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":1,\"matrix\":[[1,21],[2,42]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 21], [2, 42]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=1. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(1, [[0, 0]], [[21, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (1, [[0, 0]], [[21, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0097","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f32f10f901f093e857eed3b68ee211e3ddf7b2c665b86a3f933e47471d1bbca","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022097,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":2,\"matrix\":[[1,22],[2,44]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 22], [2, 44]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=2. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(2, [[0, 0]], [[22, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (2, [[0, 0]], [[22, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0098","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecf3f8a5d74bf42db85dc6518f446dcbeb5da1b4c4abe63901084606fa10b8f6","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022098,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":3,\"matrix\":[[1,23],[2,46]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 23], [2, 46]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=3. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(3, [[0, 0]], [[23, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (3, [[0, 0]], [[23, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0099","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd46e1588cba5ca2c368189a0acfa3c41efacf8e0215b761396dc4f5276ca1e6","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022099,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":4,\"matrix\":[[1,24],[2,48]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 24], [2, 48]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=4. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(4, [[0, 0]], [[24, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (4, [[0, 0]], [[24, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r02-linear-map-kernel-0100","batch_id":"batch-20261014-advance-20261102-r02-linear-map-kernel-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87894ffc43deb2ac1f25e76a456c7be08f35a52f8635a532460d92c36dd6a39b","family_id":"qf-mod33-kernel-bounded-search-boundary","seed":202611022100,"semantic_template":"qf-mod33-kernel-bounded-search-boundary/{\"bound\":5,\"matrix\":[[1,25],[2,50]]}"},"fields":{"title":"A Bounded Integer Box Is Not the Full Real Kernel","stem":"Use this same exact integer matrix A=[[1, 25], [2, 50]] for the usual linear map T:R²→R², with rows and columns in the supplied order. Kernel means all real x solving A*x=0; image means the full real column span, not automatically the whole codomain. An empty basis spans the zero space. Rank2 has zero-dimensional kernel and both original columns as an image basis; rank0 has the standard kernel basis and empty image basis. At rank1 the kernel uses a primitive integer normal-equation solution, with its first nonzero coordinate positive, and the image uses the first nonzero original column. All divisions are exact rational operations. Nonnegative integer bound=5. Search x,y in [−bound,bound]∩Z, with x outer increasing and y inner increasing. Give (bound,integer_kernel_points,global_kernel_basis,complete?,bounded_integer_search_not_full_real_kernel).","answer":{"kind":"expression","value":"(5, [[0, 0]], [[25, -1]], No, bounded_integer_search_not_full_real_kernel)","explanation":"List exactly the zero-target points in this box and retain the separately derived global basis. The No complete flag concerns the bounded enumeration as a general full-real certificate, not denial of the symbolic basis already supplied. The exact result, retaining the requested order, is (5, [[0, 0]], [[25, -1]], No, bounded_integer_search_not_full_real_kernel)."},"skills":["A Bounded Integer Box Is Not the Full Real Kernel"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}