Question 13051: The Matrix Formula Preserves a Whole Linear Combination

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

← Question derivation index · Original question batch

R0025-13051 · q-20261014-advance-20261027-r05-general-linear-map-0076

Original question and givens

Use the complete rectangular matrix A=[[-12, -3, -5], [-2, 12, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -2, -3], v=[-5, 1, -2], alpha=-2, beta=-3; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).

G001 — Use the complete rectangular matrix A=[[-12, -3, -5], [-2, 12, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -2, -3], v=[-5, 1, -2], alpha=-2, beta=-3; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_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-general-linear-map

Step-by-step derivation

S001 · calculation

Before

input combination coordinate0 first scaled := mul(-2, 1)

After

input combination coordinate0 first scaled = -2

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001

S002 · calculation

Before

input combination coordinate0 second scaled := mul(-3, -5)

After

input combination coordinate0 second scaled = 15

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

input combination coordinate0 sum := sum(-2, 15)

After

input combination coordinate0 sum = 13

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

input combination coordinate1 first scaled := mul(-2, -2)

After

input combination coordinate1 first scaled = 4

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

input combination coordinate1 second scaled := mul(-3, 1)

After

input combination coordinate1 second scaled = -3

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

input combination coordinate1 sum := sum(4, -3)

After

input combination coordinate1 sum = 1

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

input combination coordinate2 first scaled := mul(-2, -3)

After

input combination coordinate2 first scaled = 6

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

input combination coordinate2 second scaled := mul(-3, -2)

After

input combination coordinate2 second scaled = 6

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

input combination coordinate2 sum := sum(6, 6)

After

input combination coordinate2 sum = 12

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

combined image:matrix=[[-12, -3, -5], [-2, 12, -6]];input=[13, 1, 12]

After

R^3->R^2;each output is its original row dot input

Conditions: none

Input dimension is the column count and output dimension the row count.The matrix may be rectangular;do not transpose it or invent coordinates.

Step identifiers and dependencies

Rule: rectangular_linear_map_source_target_roles · Uses: G001, S009

S011 · calculation

Before

combined image row0 column0 term := mul(-12, 13)

After

combined image row0 column0 term = -156

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

combined image row0 column1 term := mul(-3, 1)

After

combined image row0 column1 term = -3

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

combined image row0 column2 term := mul(-5, 12)

After

combined image row0 column2 term = -60

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

combined image row0 sum := sum(-156, -3, -60)

After

combined image row0 sum = -219

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

combined image row1 column0 term := mul(-2, 13)

After

combined image row1 column0 term = -26

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

combined image row1 column1 term := mul(12, 1)

After

combined image row1 column1 term = 12

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

combined image row1 column2 term := mul(-6, 12)

After

combined image row1 column2 term = -72

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

combined image row1 sum := sum(-26, 12, -72)

After

combined image row1 sum = -86

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

u image:matrix=[[-12, -3, -5], [-2, 12, -6]];input=[1, -2, -3]

After

R^3->R^2;each output is its original row dot input

Conditions: none

Input dimension is the column count and output dimension the row count.The matrix may be rectangular;do not transpose it or invent coordinates.

Step identifiers and dependencies

Rule: rectangular_linear_map_source_target_roles · Uses: G001, S018

S020 · calculation

Before

u image row0 column0 term := mul(-12, 1)

After

u image row0 column0 term = -12

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

u image row0 column1 term := mul(-3, -2)

After

u image row0 column1 term = 6

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

u image row0 column2 term := mul(-5, -3)

After

u image row0 column2 term = 15

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

u image row0 sum := sum(-12, 6, 15)

After

u image row0 sum = 9

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

u image row1 column0 term := mul(-2, 1)

After

u image row1 column0 term = -2

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

u image row1 column1 term := mul(12, -2)

After

u image row1 column1 term = -24

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

u image row1 column2 term := mul(-6, -3)

After

u image row1 column2 term = 18

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

u image row1 sum := sum(-2, -24, 18)

After

u image row1 sum = -8

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

v image:matrix=[[-12, -3, -5], [-2, 12, -6]];input=[-5, 1, -2]

After

R^3->R^2;each output is its original row dot input

Conditions: none

Input dimension is the column count and output dimension the row count.The matrix may be rectangular;do not transpose it or invent coordinates.

Step identifiers and dependencies

Rule: rectangular_linear_map_source_target_roles · Uses: G001, S027

S029 · calculation

Before

v image row0 column0 term := mul(-12, -5)

After

v image row0 column0 term = 60

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

v image row0 column1 term := mul(-3, 1)

After

v image row0 column1 term = -3

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

v image row0 column2 term := mul(-5, -2)

After

v image row0 column2 term = 10

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

v image row0 sum := sum(60, -3, 10)

After

v image row0 sum = 67

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

v image row1 column0 term := mul(-2, -5)

After

v image row1 column0 term = 10

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

v image row1 column1 term := mul(12, 1)

After

v image row1 column1 term = 12

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

v image row1 column2 term := mul(-6, -2)

After

v image row1 column2 term = 12

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

v image row1 sum := sum(10, 12, 12)

After

v image row1 sum = 34

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

output combination coordinate0 first scaled := mul(-2, 9)

After

output combination coordinate0 first scaled = -18

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

output combination coordinate0 second scaled := mul(-3, 67)

After

output combination coordinate0 second scaled = -201

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

output combination coordinate0 sum := sum(-18, -201)

After

output combination coordinate0 sum = -219

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

output combination coordinate1 first scaled := mul(-2, -8)

After

output combination coordinate1 first scaled = 16

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

output combination coordinate1 second scaled := mul(-3, 34)

After

output combination coordinate1 second scaled = -102

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

output combination coordinate1 sum := sum(16, -102)

After

output combination coordinate1 sum = -86

Conditions: none

Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · implies

Before

matrix formula;alpha=-2;beta=-3;lhs=[-219, -86];rhs=[-219, -86]

After

all row dot products distribute over all real linear combinations;instance equality=Yes

Conditions: none

Every output coordinate is a finite sum distributing over arbitrary real scalars.All rows together give a universal linearity certificate;the full coordinate calculation also checks this instance.

Step identifiers and dependencies

Rule: matrix_combination_universal_coordinate_certificate · Uses: G001, S042

Reverse checks

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 output combination coordinate1 sum with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute output combination coordinate1 second scaled with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute output combination coordinate1 first scaled with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Recompute output combination coordinate0 sum with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Recompute output combination coordinate0 second scaled with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute output combination coordinate0 first scaled with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute v image row1 sum with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute v image row1 column2 term with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute v image row1 column1 term with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute v image row1 column0 term with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute v image row0 sum with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute v image row0 column2 term with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute v image row0 column1 term with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute v image row0 column0 term with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S027 · premise_and_result_audit — Recompute u image row1 sum with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute u image row1 column2 term with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute u image row1 column1 term with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute u image row1 column0 term with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute u image row0 sum with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute u image row0 column2 term with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute u image row0 column1 term with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Recompute u image row0 column0 term with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S018 · premise_and_result_audit — Recompute combined image row1 sum with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute combined image row1 column2 term with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute combined image row1 column1 term with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute combined image row1 column0 term with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute combined image row0 sum with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute combined image row0 column2 term with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute combined image row0 column1 term with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute combined image row0 column0 term 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 input combination coordinate2 sum with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute input combination coordinate2 second scaled with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute input combination coordinate2 first scaled with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute input combination coordinate1 sum with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute input combination coordinate1 second scaled with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute input combination coordinate1 first scaled with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute input combination coordinate0 sum with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute input combination coordinate0 second scaled with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Recompute input combination coordinate0 first scaled with exact rational arithmetic, not rounded decimals.

Result

(-2, -3, [13, 1, 12], [-219, -86], [-219, -86], Yes, matrix_formula_universal_certificate)

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: 72e137addba4f9cf588e69c34697cfdc16a6b0a0b89e2689e963f12d63c80b61
candidate record SHA-256: e505210694f72f7d0887536e2441480cc9908784f4321d8bf7a118eeaafe9e14
semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5
question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a
Lean: NOT_RUN · human review: NOT_PERFORMED

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