# Zero-Target Kernels, Full Real Bases and Bounded-Search Boundaries

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r02-linear-map-kernel-0100/)

Kernel of a linear map · qf-mod33-kernel-bounded-search-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261102-r02-linear-map-kernel-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261102-r02-linear-map-kernel-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261102-r02-linear-map-kernel-foundations/validation-report.json)