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Question 13062: The Matrix Formula Preserves a Whole Linear Combination

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0025-13062 · q-20261014-advance-20261027-r05-general-linear-map-0087

Original question and givens

Use the complete rectangular matrix A=[[-1, 1, -5], [-1, 1, -3]] 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, -1, 1], v=[-5, 1, 1], alpha=-1, beta=1; 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=[[-1, 1, -5], [-1, 1, -3]] 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, -1, 1], v=[-5, 1, 1], alpha=-1, beta=1; 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.

Step-by-step derivation

S001 · calculation

Before

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

After

input combination coordinate0 first scaled = -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

S002 · calculation

Before

input combination coordinate0 second scaled := mul(1, -5)

After

input combination coordinate0 second scaled = -5

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(-1, -5)

After

input combination coordinate0 sum = -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, S002

S004 · calculation

Before

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

After

input combination coordinate1 first scaled = 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, S003

S005 · calculation

Before

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

After

input combination coordinate1 second scaled = 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, S004

S006 · calculation

Before

input combination coordinate1 sum := sum(1, 1)

After

input combination coordinate1 sum = 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, S005

S007 · calculation

Before

input combination coordinate2 first scaled := mul(-1, 1)

After

input combination coordinate2 first scaled = -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, S006

S008 · calculation

Before

input combination coordinate2 second scaled := mul(1, 1)

After

input combination coordinate2 second scaled = 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, S007

S009 · calculation

Before

input combination coordinate2 sum := sum(-1, 1)

After

input combination coordinate2 sum = 0

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=[[-1, 1, -5], [-1, 1, -3]];input=[-6, 2, 0]

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(-1, -6)

After

combined image row0 column0 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, S010

S012 · calculation

Before

combined image row0 column1 term := mul(1, 2)

After

combined image row0 column1 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, S011

S013 · calculation

Before

combined image row0 column2 term := mul(-5, 0)

After

combined image row0 column2 term = 0

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(6, 2, 0)

After

combined image row0 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, S013

S015 · calculation

Before

combined image row1 column0 term := mul(-1, -6)

After

combined image row1 column0 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, S014

S016 · calculation

Before

combined image row1 column1 term := mul(1, 2)

After

combined image row1 column1 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, S015

S017 · calculation

Before

combined image row1 column2 term := mul(-3, 0)

After

combined image row1 column2 term = 0

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(6, 2, 0)

After

combined 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, S017

S019 · implies

Before

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

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(-1, 1)

After

u image row0 column0 term = -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, S019

S021 · calculation

Before

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

After

u image row0 column1 term = -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, S020

S022 · calculation

Before

u image row0 column2 term := mul(-5, 1)

After

u image row0 column2 term = -5

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(-1, -1, -5)

After

u image row0 sum = -7

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(-1, 1)

After

u image row1 column0 term = -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, S023

S025 · calculation

Before

u image row1 column1 term := mul(1, -1)

After

u image row1 column1 term = -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, S024

S026 · calculation

Before

u image row1 column2 term := mul(-3, 1)

After

u image row1 column2 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, S025

S027 · calculation

Before

u image row1 sum := sum(-1, -1, -3)

After

u image row1 sum = -5

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=[[-1, 1, -5], [-1, 1, -3]];input=[-5, 1, 1]

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(-1, -5)

After

v image row0 column0 term = 5

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(1, 1)

After

v image row0 column1 term = 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, S029

S031 · calculation

Before

v image row0 column2 term := mul(-5, 1)

After

v image row0 column2 term = -5

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(5, 1, -5)

After

v image row0 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, S031

S033 · calculation

Before

v image row1 column0 term := mul(-1, -5)

After

v image row1 column0 term = 5

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(1, 1)

After

v image row1 column1 term = 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, S033

S035 · calculation

Before

v image row1 column2 term := mul(-3, 1)

After

v image row1 column2 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, S034

S036 · calculation

Before

v image row1 sum := sum(5, 1, -3)

After

v image row1 sum = 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, S035

S037 · calculation

Before

output combination coordinate0 first scaled := mul(-1, -7)

After

output combination coordinate0 first scaled = 7

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(1, 1)

After

output combination coordinate0 second scaled = 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, S037

S039 · calculation

Before

output combination coordinate0 sum := sum(7, 1)

After

output combination coordinate0 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, S038

S040 · calculation

Before

output combination coordinate1 first scaled := mul(-1, -5)

After

output combination coordinate1 first scaled = 5

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(1, 3)

After

output 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, S040

S042 · calculation

Before

output combination coordinate1 sum := sum(5, 3)

After

output combination coordinate1 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, S041

S043 · implies

Before

matrix formula;alpha=-1;beta=1;lhs=[8, 8];rhs=[8, 8]

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

(-1, 1, [-6, 2, 0], [8, 8], [8, 8], 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: fb9c899242db6d36d908560c709becdd70bd6cfad39a77cafc28d23f2b34c1be
  • candidate record SHA-256: 194aacc5516850a351a21cc15672470d505efddb6c85252af756e31d1495f4c0
  • semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5
  • question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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