# 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

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

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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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0100/)

Rank-nullity theorem · qf-mod35-ranknull-sampled-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations/validation-report.json)