# Question 18313: The Source Dimension Equals Nullity Plus Rank [Return to catalogue page 92](https://ocme.evemisslab.com/en/questions/page/92/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261104-r01-rank-nullity-theorem-foundations/#q-20261014-advance-20261104-r01-rank-nullity-theorem-0038) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0035-18313 · q-20261014-advance-20261104-r01-rank-nullity-theorem-0038 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-rank-nullity-theorem](https://ocme.evemisslab.com/en/mko/mko-rank-nullity-theorem/) ## Step-by-step derivation S001 · impliesBefore source operation=ranknull-dimension-identity;original snapshot and mathematics retained After explicit ordered source output fields=[source_dimension, nullity, rank, sum, equal] Conditions: none Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore complete original matrix=[[1, 0, 12], [0, 1, 1]];shape=2 rows by3 columns After usual real map T:R^3 to R^2;SOURCE dimension=3;TARGET dimension=2 Conditions: none Rows give target dimension and columns source dimension.Do not force this rectangular map into a square or earlier R²-only case. Step identifiers and dependenciesRule: ranknull_rectangular_map_direction_source_columns · Uses: G001, S001 S003 · calculationBefore ranknull pivot0 column0 normalized entry0 := div(1, 1) After ranknull pivot0 column0 normalized entry0 = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore ranknull pivot0 column0 normalized entry1 := div(0, 1) After ranknull pivot0 column0 normalized entry1 = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore ranknull pivot0 column0 normalized entry2 := div(12, 1) After ranknull pivot0 column0 normalized entry2 = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore pivot row=0;column=0;nonzero exact pivot=1;old row=[1, 0, 12] After normalized complete row=[1, 0, 12] Conditions: none Divide the entire row by its nonzero pivot with exact rational values,not floats. Step identifiers and dependenciesRule: ranknull_exact_pivot_scaling_all_entries · Uses: G001, S005 S007 · calculationBefore ranknull pivot1 column1 normalized entry0 := div(0, 1) After ranknull pivot1 column1 normalized entry0 = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore ranknull pivot1 column1 normalized entry1 := div(1, 1) After ranknull pivot1 column1 normalized entry1 = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore ranknull pivot1 column1 normalized entry2 := div(1, 1) After ranknull pivot1 column1 normalized entry2 = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore pivot row=1;column=1;nonzero exact pivot=1;old row=[0, 1, 1] After normalized complete row=[0, 1, 1] Conditions: none Divide the entire row by its nonzero pivot with exact rational values,not floats. Step identifiers and dependenciesRule: ranknull_exact_pivot_scaling_all_entries · Uses: G001, S009 S011 · calculationBefore ranknull free2 pivot0 solved coordinate := mul(-1, 12) After ranknull free2 pivot0 solved coordinate = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore ranknull free2 pivot1 solved coordinate := mul(-1, 1) After ranknull free2 pivot1 solved coordinate = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore selected free column=2;ALL free columns=[2];same exact RREF=[[1, 0, 12], [0, 1, 1]] After set selected free coordinate1,other free coordinates0;solved kernel vector=[-12, -1, 1];NO primitive/sign renormalization Conditions: none One basis vector per free column sets that free coordinate to1 and the others to0,then solves pivot coordinates.Keep this RREF convention rather than the earlier primitive-integer positive-leading convention. Step identifiers and dependenciesRule: ranknull_kernel_free_coordinate_source_convention · Uses: G001, S012 S014 · calculationBefore ranknull kernel free2 original row0 term0 := mul(1, -12) After ranknull kernel free2 original row0 term0 = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore ranknull kernel free2 original row0 term1 := mul(0, -1) After ranknull kernel free2 original row0 term1 = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore ranknull kernel free2 original row0 term2 := mul(12, 1) After ranknull kernel free2 original row0 term2 = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore ranknull kernel free2 original row0 sum := sum(-12, 0, 12) After ranknull kernel free2 original row0 sum = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore ranknull kernel free2 original row1 term0 := mul(0, -12) After ranknull kernel free2 original row1 term0 = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore ranknull kernel free2 original row1 term1 := mul(1, -1) After ranknull kernel free2 original row1 term1 = -1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore ranknull kernel free2 original row1 term2 := mul(1, 1) After ranknull kernel free2 original row1 term2 = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore ranknull kernel free2 original row1 sum := sum(0, -1, 1) After ranknull kernel free2 original row1 sum = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · impliesBefore complete exact RREF=[[1, 0, 12], [0, 1, 1]];pivot columns=[0, 1];free columns=[2];verified kernel vectors=[[-12, -1, 1]] After for ANY real solution,its free coordinates are unique combination coefficients of these vectors;pivot equations determine the rest;free-coordinate identity also proves independence Conditions: none Invertible row operations preserve Mx=0.Every solution's free coordinates are the unique coefficients in this family;the pivot equations determine the rest.Conversely every combination maps to zero.The free-coordinate identity proves independence,so this is a full basis,not samples. Step identifiers and dependenciesRule: ranknull_full_real_kernel_both_inclusions_and_independence · Uses: G001, S021 S023 · calculationBefore ranknull image column0 row0 pivot0 term := mul(1, 1) After ranknull image column0 row0 pivot0 term = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore ranknull image column0 row0 pivot1 term := mul(0, 0) After ranknull image column0 row0 pivot1 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore ranknull image column0 row0 sum := sum(1, 0) After ranknull image column0 row0 sum = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore ranknull image column0 row1 pivot0 term := mul(0, 1) After ranknull image column0 row1 pivot0 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore ranknull image column0 row1 pivot1 term := mul(1, 0) After ranknull image column0 row1 pivot1 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore ranknull image column0 row1 sum := sum(0, 0) After ranknull image column0 row1 sum = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore original column index=0;ORIGINAL pivot-column basis=[[1, 0], [0, 1]];RREF coordinate coefficients=[1, 0] After reconstructed original column=[1, 0] Conditions: none Reconstruct every original column from the original pivot columns,so the full image lies in their span.Each pivot column itself equals M e_p,giving the reverse inclusion.Do not substitute RREF columns as the original image basis. Step identifiers and dependenciesRule: ranknull_original_pivot_columns_not_rref_image_columns · Uses: G001, S028 S030 · calculationBefore ranknull image column1 row0 pivot0 term := mul(1, 0) After ranknull image column1 row0 pivot0 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore ranknull image column1 row0 pivot1 term := mul(0, 1) After ranknull image column1 row0 pivot1 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore ranknull image column1 row0 sum := sum(0, 0) After ranknull image column1 row0 sum = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore ranknull image column1 row1 pivot0 term := mul(0, 0) After ranknull image column1 row1 pivot0 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore ranknull image column1 row1 pivot1 term := mul(1, 1) After ranknull image column1 row1 pivot1 term = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore ranknull image column1 row1 sum := sum(0, 1) After ranknull image column1 row1 sum = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · impliesBefore original column index=1;ORIGINAL pivot-column basis=[[1, 0], [0, 1]];RREF coordinate coefficients=[0, 1] After reconstructed original column=[0, 1] Conditions: none Reconstruct every original column from the original pivot columns,so the full image lies in their span.Each pivot column itself equals M e_p,giving the reverse inclusion.Do not substitute RREF columns as the original image basis. Step identifiers and dependenciesRule: ranknull_original_pivot_columns_not_rref_image_columns · Uses: G001, S035 S037 · calculationBefore ranknull image column2 row0 pivot0 term := mul(1, 12) After ranknull image column2 row0 pivot0 term = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore ranknull image column2 row0 pivot1 term := mul(0, 1) After ranknull image column2 row0 pivot1 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore ranknull image column2 row0 sum := sum(12, 0) After ranknull image column2 row0 sum = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore ranknull image column2 row1 pivot0 term := mul(0, 12) After ranknull image column2 row1 pivot0 term = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore ranknull image column2 row1 pivot1 term := mul(1, 1) After ranknull image column2 row1 pivot1 term = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore ranknull image column2 row1 sum := sum(0, 1) After ranknull image column2 row1 sum = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · impliesBefore original column index=2;ORIGINAL pivot-column basis=[[1, 0], [0, 1]];RREF coordinate coefficients=[12, 1] After reconstructed original column=[12, 1] Conditions: none Reconstruct every original column from the original pivot columns,so the full image lies in their span.Each pivot column itself equals M e_p,giving the reverse inclusion.Do not substitute RREF columns as the original image basis. Step identifiers and dependenciesRule: ranknull_original_pivot_columns_not_rref_image_columns · Uses: G001, S042 S044 · impliesBefore ORIGINAL pivot-column family=[[1, 0], [0, 1]];pivot indices=[0, 1];same RREF pivot columns=[[1, 0], [0, 1]] After ANY zero combination stays zero under invertible row operations;distinct target-coordinate identity columns force EVERY coefficient0;family independent,and previous reconstructions prove full image span Conditions: none Apply the same invertible row operations to any zero combination of original pivot columns.The resulting distinct target-coordinate identity columns force all coefficients to zero.With the full-span reconstructions this is an image basis.The empty family is independent and spans the zero space. Step identifiers and dependenciesRule: ranknull_image_basis_independence_via_pivot_identity · Uses: G001, S043 S045 · calculationBefore ranknull source dimension identity sum := sum(1, 2) After ranknull source dimension identity sum = 3 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · impliesBefore original matrix shape=2x3;rank=2;FULL kernel basis count=1;FULL original-pivot image basis count=2 After SOURCE column dimension=3=nullity+rank=3;target row dimension=2 retained separately Conditions: none Pivot/free columns partition source coordinates,and the full kernel/image bases give nullity and rank.Nullity is not the number of solutions or grid points,and target rows do not replace source columns. Step identifiers and dependenciesRule: ranknull_dimension_identity_not_solution_count · Uses: G001, S045 ## Reverse checks - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Recompute ranknull source dimension identity sum with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Recompute ranknull image column2 row1 sum with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute ranknull image column2 row1 pivot1 term with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute ranknull image column2 row1 pivot0 term with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute ranknull image column2 row0 sum with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute ranknull image column2 row0 pivot1 term with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute ranknull image column2 row0 pivot0 term with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Recompute ranknull image column1 row1 sum with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute ranknull image column1 row1 pivot1 term with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute ranknull image column1 row1 pivot0 term with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute ranknull image column1 row0 sum with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute ranknull image column1 row0 pivot1 term with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute ranknull image column1 row0 pivot0 term with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Recompute ranknull image column0 row1 sum with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute ranknull image column0 row1 pivot1 term with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute ranknull image column0 row1 pivot0 term with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute ranknull image column0 row0 sum with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute ranknull image column0 row0 pivot1 term with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute ranknull image column0 row0 pivot0 term with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Recompute ranknull kernel free2 original row1 sum with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute ranknull kernel free2 original row1 term2 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute ranknull kernel free2 original row1 term1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute ranknull kernel free2 original row1 term0 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute ranknull kernel free2 original row0 sum with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute ranknull kernel free2 original row0 term2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute ranknull kernel free2 original row0 term1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute ranknull kernel free2 original row0 term0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute ranknull free2 pivot1 solved coordinate with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute ranknull free2 pivot0 solved coordinate with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute ranknull pivot1 column1 normalized entry2 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute ranknull pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute ranknull pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute ranknull pivot0 column0 normalized entry2 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute ranknull pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute ranknull pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (3, 1, 2, 3, Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 67be19b101f3db22e825eeaa83007990651e55548f5491809cd2df31585e1c84 - candidate record SHA-256: 9377bacfb7b72a66e5306e2fed006a19d326b7a68e4226b9dea8f16bdb042737 - semantic review receipt SHA-256: 713db6309b22bf8b31f0253d4aaa7aca9011ab908587ae0e9d4a365edce1f9bc - question-first witness SHA-256: 0338a3db344ab237113fc63be29153c3684da96e388109e9dcd913a0a693aed2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261104-r01-rank-nullity-theorem-0038.json)