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Full Real Column Images, Exact Membership and Noninteger Preimages

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batch-20261014-advance-20261102-r03-linear-map-image-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. Image Membership Uses the Full Real Column Span

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. Target=[2, 4]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([2, 4], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([2, 4], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

2. Image Membership Uses the Full Real Column Span

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. Target=[3, 6]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([3, 6], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([3, 6], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

3. Image Membership Uses the Full Real Column Span

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. Target=[4, 8]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([4, 8], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([4, 8], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

4. Image Membership Uses the Full Real Column Span

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. Target=[5, 10]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([5, 10], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([5, 10], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

5. Image Membership Uses the Full Real Column Span

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. Target=[6, 12]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([6, 12], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([6, 12], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

6. Image Membership Uses the Full Real Column Span

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. Target=[7, 14]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([7, 14], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([7, 14], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

7. Image Membership Uses the Full Real Column Span

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. Target=[8, 16]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([8, 16], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([8, 16], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

8. Image Membership Uses the Full Real Column Span

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. Target=[9, 18]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([9, 18], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([9, 18], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

9. Image Membership Uses the Full Real Column Span

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. Target=[10, 20]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([10, 20], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([10, 20], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

10. Image Membership Uses the Full Real Column Span

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. Target=[11, 22]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([11, 22], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([11, 22], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

11. Image Membership Uses the Full Real Column Span

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. Target=[12, 24]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([12, 24], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([12, 24], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

12. Image Membership Uses the Full Real Column Span

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. Target=[13, 26]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([13, 26], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([13, 26], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

13. Image Membership Uses the Full Real Column Span

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. Target=[14, 28]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([14, 28], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([14, 28], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

14. Image Membership Uses the Full Real Column Span

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. Target=[15, 30]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([15, 30], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([15, 30], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

15. Image Membership Uses the Full Real Column Span

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. Target=[16, 32]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([16, 32], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([16, 32], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

16. Image Membership Uses the Full Real Column Span

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. Target=[17, 34]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([17, 34], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([17, 34], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

17. Image Membership Uses the Full Real Column Span

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. Target=[18, 36]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([18, 36], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([18, 36], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

18. Image Membership Uses the Full Real Column Span

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. Target=[19, 38]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([19, 38], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([19, 38], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

19. Image Membership Uses the Full Real Column Span

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. Target=[20, 40]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([20, 40], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([20, 40], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

20. Image Membership Uses the Full Real Column Span

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. Target=[21, 42]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([21, 42], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([21, 42], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

21. Image Membership Uses the Full Real Column Span

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. Target=[22, 44]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([22, 44], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([22, 44], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

22. Image Membership Uses the Full Real Column Span

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. Target=[23, 46]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([23, 46], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([23, 46], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

23. Image Membership Uses the Full Real Column Span

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. Target=[24, 48]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([24, 48], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([24, 48], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

24. Image Membership Uses the Full Real Column Span

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. Target=[25, 50]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([25, 50], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([25, 50], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

25. Image Membership Uses the Full Real Column Span

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. Target=[26, 52]. Give (target,rank,image_basis,in_image?,certificate). A rank2 certificate is [unique_solution,exact_coordinates]; rank0 uses zero_image or nonzero_target_zero_image; rank1 uses [column_span_cross,basis_vector,cross_value].

View answer and explanation

([26, 52], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0])

At full rank solve A*x=target exactly, allowing noninteger rational coordinates. At rank1 test collinearity with the nonzero column; at rank0 only the zero target is in the image. These are full-real consistency checks, not an integer-only search. The exact result, retaining the requested order, is ([26, 52], 1, [[1, 2]], Yes, [column_span_cross, [1, 2], 0]).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-membership · candidate

26. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 1], [2, 2]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 1], [2, 2]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

27. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 2], [2, 4]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 2], [2, 4]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

28. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 3], [2, 6]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 3], [2, 6]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

29. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 4], [2, 8]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 4], [2, 8]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

30. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 5], [2, 10]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 5], [2, 10]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

31. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 6], [2, 12]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 6], [2, 12]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

32. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 7], [2, 14]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 7], [2, 14]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

33. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 8], [2, 16]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 8], [2, 16]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

34. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 9], [2, 18]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 9], [2, 18]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

35. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 10], [2, 20]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 10], [2, 20]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

36. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 11], [2, 22]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 11], [2, 22]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

37. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 12], [2, 24]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 12], [2, 24]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

38. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 13], [2, 26]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 13], [2, 26]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

39. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 14], [2, 28]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 14], [2, 28]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

40. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 15], [2, 30]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 15], [2, 30]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

41. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 16], [2, 32]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 16], [2, 32]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

42. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 17], [2, 34]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 17], [2, 34]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

43. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 18], [2, 36]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 18], [2, 36]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

44. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 19], [2, 38]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 19], [2, 38]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

45. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 20], [2, 40]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 20], [2, 40]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

46. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 21], [2, 42]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 21], [2, 42]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

47. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 22], [2, 44]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 22], [2, 44]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

48. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 23], [2, 46]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 23], [2, 46]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

49. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 24], [2, 48]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 24], [2, 48]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

50. The Image Basis Comes from Original Matrix Columns

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,image_basis,full_real_column_span).

View answer and explanation

([[1, 25], [2, 50]], 1, [[1, 2]], full_real_column_span)

Select the stated original column basis and take its full real span. Do not substitute rows, assume the image equals the codomain, or normalize columns into different output vectors. The exact result, retaining the requested order, is ([[1, 25], [2, 50]], 1, [[1, 2]], full_real_column_span).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-basis · candidate

51. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

52. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

53. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

54. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

55. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

56. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

57. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

58. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

59. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

60. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

61. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

62. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

63. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

64. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

65. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

66. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

67. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

68. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

69. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

70. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

71. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

72. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

73. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

74. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

75. Rank Is the Dimension of the Full Real Image

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,basis_count,equal?).

View answer and explanation

(1, 1, Yes)

Count the supplied full image basis and compare with exact matrix rank. A rank0 map has an empty image basis even though its image contains the zero vector. The exact result, retaining the requested order, is (1, 1, Yes).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-rank · candidate

76. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[1, 1], nonnegative bound=1. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([1, 1], 1, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([1, 1], 1, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

77. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[3, 1], nonnegative bound=2. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([3, 1], 2, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([3, 1], 2, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

78. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[5, 1], nonnegative bound=3. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([5, 1], 3, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([5, 1], 3, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

79. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[7, 1], nonnegative bound=4. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([7, 1], 4, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([7, 1], 4, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

80. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[9, 1], nonnegative bound=5. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([9, 1], 5, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([9, 1], 5, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

81. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[11, 1], nonnegative bound=1. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([11, 1], 1, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([11, 1], 1, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

82. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[13, 1], nonnegative bound=2. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([13, 1], 2, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([13, 1], 2, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

83. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[15, 1], nonnegative bound=3. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([15, 1], 3, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([15, 1], 3, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

84. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[17, 1], nonnegative bound=4. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([17, 1], 4, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([17, 1], 4, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

85. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[19, 1], nonnegative bound=5. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([19, 1], 5, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([19, 1], 5, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

86. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[21, 1], nonnegative bound=1. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([21, 1], 1, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([21, 1], 1, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

87. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[23, 1], nonnegative bound=2. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([23, 1], 2, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([23, 1], 2, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

88. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[25, 1], nonnegative bound=3. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([25, 1], 3, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([25, 1], 3, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

89. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[27, 1], nonnegative bound=4. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([27, 1], 4, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([27, 1], 4, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

90. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[29, 1], nonnegative bound=5. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([29, 1], 5, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([29, 1], 5, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

91. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[31, 1], nonnegative bound=1. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([31, 1], 1, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([31, 1], 1, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

92. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[33, 1], nonnegative bound=2. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([33, 1], 2, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([33, 1], 2, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

93. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[35, 1], nonnegative bound=3. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([35, 1], 3, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([35, 1], 3, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

94. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[37, 1], nonnegative bound=4. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([37, 1], 4, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([37, 1], 4, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

95. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[39, 1], nonnegative bound=5. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([39, 1], 5, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([39, 1], 5, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

96. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[41, 1], nonnegative bound=1. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([41, 1], 1, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([41, 1], 1, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

97. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[43, 1], nonnegative bound=2. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([43, 1], 2, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([43, 1], 2, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

98. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[45, 1], nonnegative bound=3. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([45, 1], 3, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([45, 1], 3, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

99. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[47, 1], nonnegative bound=4. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([47, 1], 4, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([47, 1], 4, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

100. Failure to Find an Integer Preimage Does Not Establish Real Nonmembership

Use this same exact integer matrix A=[[2, 0], [0, 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. Target=[49, 1], nonnegative bound=5. Search integer pairs x,y in [−bound,bound] in x-then-y increasing order. Give (target,bound,integer_witnesses,global_membership?,boundary).

View answer and explanation

([49, 1], 5, [], Yes, bounded_failure_not_global_nonmembership)

List only preimages in this bounded integer box, then separately test full-real column-span membership. No integer witness together with true real membership yields bounded_failure_not_global_nonmembership; otherwise retain bounded_search_only. A valid noninteger solution can be missed by every integer box. The exact result, retaining the requested order, is ([49, 1], 5, [], Yes, bounded_failure_not_global_nonmembership).

Reviewed step-by-step derivation

Image of a linear map · qf-mod33-image-bounded-search-boundary · candidate

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