Question 18333: Use the Full Kernel and Original Pivot Columns

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0035-18333 · q-20261014-advance-20261104-r01-rank-nullity-theorem-0058

Original question and givens

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).

G001 — 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).

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

Step-by-step derivation

S001 · implies

Before

source operation=ranknull-kernel-image-certificate;original snapshot and mathematics retained

After

explicit ordered source output fields=[kernel_basis, original_pivot_column_basis, nullity, rank, identity, certificate]

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 dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

complete original matrix=[[1, 0, 7], [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 dependencies

Rule: ranknull_rectangular_map_direction_source_columns · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

ranknull pivot0 column0 normalized entry2 := div(7, 1)

After

ranknull pivot0 column0 normalized entry2 = 7

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

pivot row=0;column=0;nonzero exact pivot=1;old row=[1, 0, 7]

After

normalized complete row=[1, 0, 7]

Conditions: none

Divide the entire row by its nonzero pivot with exact rational values,not floats.

Step identifiers and dependencies

Rule: ranknull_exact_pivot_scaling_all_entries · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

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 dependencies

Rule: ranknull_exact_pivot_scaling_all_entries · Uses: G001, S009

S011 · calculation

Before

ranknull free2 pivot0 solved coordinate := mul(-1, 7)

After

ranknull free2 pivot0 solved coordinate = -7

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

selected free column=2;ALL free columns=[2];same exact RREF=[[1, 0, 7], [0, 1, 1]]

After

set selected free coordinate1,other free coordinates0;solved kernel vector=[-7, -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 dependencies

Rule: ranknull_kernel_free_coordinate_source_convention · Uses: G001, S012

S014 · calculation

Before

ranknull kernel free2 original row0 term0 := mul(1, -7)

After

ranknull kernel free2 original row0 term0 = -7

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

ranknull kernel free2 original row0 term2 := mul(7, 1)

After

ranknull kernel free2 original row0 term2 = 7

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

ranknull kernel free2 original row0 sum := sum(-7, 0, 7)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

ranknull kernel free2 original row1 term0 := mul(0, -7)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

complete exact RREF=[[1, 0, 7], [0, 1, 1]];pivot columns=[0, 1];free columns=[2];verified kernel vectors=[[-7, -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 dependencies

Rule: ranknull_full_real_kernel_both_inclusions_and_independence · Uses: G001, S021

S023 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

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 dependencies

Rule: ranknull_original_pivot_columns_not_rref_image_columns · Uses: G001, S028

S030 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

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

Step identifiers and dependencies

Rule: ranknull_original_pivot_columns_not_rref_image_columns · Uses: G001, S035

S037 · calculation

Before

ranknull image column2 row0 pivot0 term := mul(1, 7)

After

ranknull image column2 row0 pivot0 term = 7

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

ranknull image column2 row0 sum := sum(7, 0)

After

ranknull image column2 row0 sum = 7

Conditions: none

Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

ranknull image column2 row1 pivot0 term := mul(0, 7)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · implies

Before

original column index=2;ORIGINAL pivot-column basis=[[1, 0], [0, 1]];RREF coordinate coefficients=[7, 1]

After

reconstructed original column=[7, 1]

Conditions: none

Step identifiers and dependencies

Rule: ranknull_original_pivot_columns_not_rref_image_columns · Uses: G001, S042

S044 · implies

Before

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 dependencies

Rule: ranknull_image_basis_independence_via_pivot_identity · Uses: G001, S043

S045 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · implies

Before

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 dependencies

Rule: 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

([[-7, -1, 1]], [[1, 0], [0, 1]], 1, 2, Yes, exact_symbolic_certificates)

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: 33b64f6f43eac5fa34bbfffb6ee87f5feb77157774b2706a444a8c6a5da5acd4
candidate record SHA-256: 2a838349dca031a1f7aa3b67ebf22e05272f8f04d6d00cc14d062827dbeb8863
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Lean: NOT_RUN · human review: NOT_PERFORMED

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