Zero-Target Kernels, Full Real Bases and Bounded-Search Boundaries
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batch-20261014-advance-20261102-r02-linear-map-kernel-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-1, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
2. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-2, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
3. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-3, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
4. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-4, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
5. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-5, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
6. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-6, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
7. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-7, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
8. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-8, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
9. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-9, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
10. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-10, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
11. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-11, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
12. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-12, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
13. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-13, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
14. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-14, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
15. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-15, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
16. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-16, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
17. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-17, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
18. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-18, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
19. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-19, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
20. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-20, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
21. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-21, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
22. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-22, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
23. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-23, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
24. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-24, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
25. Kernel Membership Has the Zero Vector as Its Target
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).
View answer and explanation
([-25, 1], [0, 0], Yes, kernel_target_is_zero_vector)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-membership · candidate
26. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 1], [2, 2]], 1, [[1, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
27. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 2], [2, 4]], 1, [[2, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
28. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 3], [2, 6]], 1, [[3, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
29. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 4], [2, 8]], 1, [[4, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
30. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 5], [2, 10]], 1, [[5, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
31. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 6], [2, 12]], 1, [[6, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
32. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 7], [2, 14]], 1, [[7, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
33. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 8], [2, 16]], 1, [[8, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
34. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 9], [2, 18]], 1, [[9, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
35. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 10], [2, 20]], 1, [[10, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
36. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 11], [2, 22]], 1, [[11, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
37. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 12], [2, 24]], 1, [[12, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
38. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 13], [2, 26]], 1, [[13, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
39. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 14], [2, 28]], 1, [[14, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
40. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 15], [2, 30]], 1, [[15, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
41. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 16], [2, 32]], 1, [[16, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
42. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 17], [2, 34]], 1, [[17, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
43. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 18], [2, 36]], 1, [[18, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
44. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 19], [2, 38]], 1, [[19, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
45. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 20], [2, 40]], 1, [[20, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
46. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 21], [2, 42]], 1, [[21, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
47. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 22], [2, 44]], 1, [[22, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
48. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 23], [2, 46]], 1, [[23, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
49. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 24], [2, 48]], 1, [[24, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
50. Give a Basis of the Entire Real Kernel, Not a Finite Search Result
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).
View answer and explanation
([[1, 25], [2, 50]], 1, [[25, -1]], 1, symbolic_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-basis · candidate
51. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[1, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
52. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[2, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
53. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[3, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
54. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[4, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
55. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[5, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
56. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[6, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
57. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[7, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
58. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[8, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
59. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[9, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
60. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[10, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
61. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[11, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
62. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[12, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
63. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[13, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
64. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[14, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
65. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[15, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
66. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[16, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
67. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[17, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
68. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[18, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
69. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[19, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
70. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[20, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
71. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[21, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
72. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[22, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
73. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[23, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
74. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[24, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
75. Nullity Counts a Full Kernel Basis
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).
View answer and explanation
(1, 1, [[25, -1]])
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]]).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-nullity · candidate
76. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(1, [[-1, 1], [0, 0], [1, -1]], [[1, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
77. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(2, [[-2, 1], [0, 0], [2, -1]], [[2, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
78. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(3, [[-3, 1], [0, 0], [3, -1]], [[3, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
79. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(4, [[-4, 1], [0, 0], [4, -1]], [[4, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
80. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(5, [[-5, 1], [0, 0], [5, -1]], [[5, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
81. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(1, [[0, 0]], [[6, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
82. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(2, [[0, 0]], [[7, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
83. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(3, [[0, 0]], [[8, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
84. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(4, [[0, 0]], [[9, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
85. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(5, [[0, 0]], [[10, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
86. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(1, [[0, 0]], [[11, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
87. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(2, [[0, 0]], [[12, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
88. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(3, [[0, 0]], [[13, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
89. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(4, [[0, 0]], [[14, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
90. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(5, [[0, 0]], [[15, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
91. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(1, [[0, 0]], [[16, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
92. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(2, [[0, 0]], [[17, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
93. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(3, [[0, 0]], [[18, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
94. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(4, [[0, 0]], [[19, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
95. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(5, [[0, 0]], [[20, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
96. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(1, [[0, 0]], [[21, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
97. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(2, [[0, 0]], [[22, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
98. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(3, [[0, 0]], [[23, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
99. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(4, [[0, 0]], [[24, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
100. A Bounded Integer Box Is Not the Full Real Kernel
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).
View answer and explanation
(5, [[0, 0]], [[25, -1]], No, bounded_integer_search_not_full_real_kernel)
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).
Reviewed step-by-step derivation
Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate
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