Rectangular Map Direction, Exact Kernel/Image Bases and Rank-Nullity
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batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations · 2026-10-14 · 100 questions
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1. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 0], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 0], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 0], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
2. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 1], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 1], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 1], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
3. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 2], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 2], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 2], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
4. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 3], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 3], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 3], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
5. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 4], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 4], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 4], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
6. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 5], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 5], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 5], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
7. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 6], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 6], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 6], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
8. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 7], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 7], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 7], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
9. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 8], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 8], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 8], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
10. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 9], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 9], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 9], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
11. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 10], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 10], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 10], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
12. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 11], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 11], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 11], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
13. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 12], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 12], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 12], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
14. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 13], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 13], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 13], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
15. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 14], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 14], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 14], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
16. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 15], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 15], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 15], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
17. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 16], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 16], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 16], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
18. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 17], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 17], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 17], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
19. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 18], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 18], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 18], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
20. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 19], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 19], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 19], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
21. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 20], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 20], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 20], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
22. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 21], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 21], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 21], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
23. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 22], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 22], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 22], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
24. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 23], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 23], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 23], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
25. Rectangular Rank-Nullity Starts with the Correct Map Direction
Use the complete exact matrix A=[[1, 0, 24], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (target_dimension, source_dimension, rref, pivot_columns, free_columns, rank, nullity).
View answer and explanation
(2, 3, [[1, 0, 24], [0, 1, 1]], [0, 1], [2], 2, 1)
Carry out the complete RREF and list zero-based pivot columns and free columns increasingly. Retain the rectangular row/column dimensions separately before reporting rank and nullity. The exact result, retaining the requested order, is (2, 3, [[1, 0, 24], [0, 1, 1]], [0, 1], [2], 2, 1).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-qualification · candidate
26. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 0], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
27. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 1], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
28. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 2], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
29. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 3], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
30. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 4], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
31. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 5], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
32. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 6], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
33. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 7], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
34. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 8], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
35. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 9], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
36. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 10], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
37. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 11], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
38. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 12], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
39. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 13], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
40. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 14], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
41. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 15], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
42. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 16], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
43. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 17], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
44. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 18], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
45. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 19], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
46. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 20], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
47. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 21], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
48. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 22], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
49. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 23], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
50. The Source Dimension Equals Nullity Plus Rank
Use the complete exact matrix A=[[1, 0, 24], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (source_dimension, nullity, rank, sum, equal).
View answer and explanation
(3, 1, 2, 3, Yes)
Compare the column count with full-kernel dimension plus full-image dimension. This is a certificate for the stated finite map, not a claim that finitely sampled vectors establish the general theorem. The exact result, retaining the requested order, is (3, 1, 2, 3, Yes).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-dimension-identity · candidate
51. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 0], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[0, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[0, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
52. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 1], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-1, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-1, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
53. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 2], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-2, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-2, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
54. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 3], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-3, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-3, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
55. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 4], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-4, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-4, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
56. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 5], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-5, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-5, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
57. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 6], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-6, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-6, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
58. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 7], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-7, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-7, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
59. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 8], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-8, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-8, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
60. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 9], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-9, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-9, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
61. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 10], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-10, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-10, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
62. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 11], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-11, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-11, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
63. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 12], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-12, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-12, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
64. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 13], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-13, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-13, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
65. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 14], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-14, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-14, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
66. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 15], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-15, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-15, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
67. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 16], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-16, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-16, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
68. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 17], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-17, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-17, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
69. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 18], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-18, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-18, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
70. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 19], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-19, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-19, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
71. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 20], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-20, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-20, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
72. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 21], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-21, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-21, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
73. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 22], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-22, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-22, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
74. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 23], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-23, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-23, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
75. Use the Full Kernel and Original Pivot Columns
Use the complete exact matrix A=[[1, 0, 24], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Give (kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate).
View answer and explanation
([[-24, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)
Construct one kernel vector per free coordinate and use the original pivot columns, not RREF columns, as the image basis. Their cardinalities are nullity and rank; the final identity flag compares their sum with source dimension. The exact result, retaining the requested order, is ([[-24, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-kernel-image-certificate · candidate
76. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 0], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
77. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 1], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
78. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 2], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
79. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 3], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
80. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 4], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
81. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 5], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
82. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 6], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
83. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 7], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
84. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 8], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
85. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 9], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
86. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 10], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
87. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 11], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
88. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 12], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
89. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 13], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
90. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 14], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
91. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 15], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
92. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 16], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
93. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 17], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
94. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 18], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
95. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 19], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
96. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 20], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
97. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 21], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
98. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 22], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
99. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 23], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
100. Sampled Vectors Cannot Replace the Full Kernel/Image Certificates
Use the complete exact matrix A=[[1, 0, 24], [0, 1, 1]] of shape 2×3, defining T:R^3→R^2. Source dimension is the column count, not the row count. Exact rational RREF gives pivot/free columns, a full real kernel basis and original pivot-column image basis. For each free column set that free coordinate to 1, other free coordinates to 0, and solve pivot coordinates; retain this RREF basis convention without sign renormalization. Nullity is a dimension, not the number of solutions. Supplied finite samples=[[0, 0, 0], [1, 0, 0]]. Give (samples, sufficient, boundary).
View answer and explanation
([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity)
The No sufficiency flag withholds a theorem or full-space conclusion from these samples alone. It does not deny the separate exact symbolic certificates available from the complete matrix. The exact result, retaining the requested order, is ([[0, 0, 0], [1, 0, 0]], No, sampled_vectors_do_not_prove_kernel_image_or_rank_nullity).
Reviewed step-by-step derivation
Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate
Canonical question JSONL · English JSONL · Validation report