OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 13072: 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

Return to catalogue page 66

← Question derivation index · Original question batch

R0025-13072 · q-20261014-advance-20261027-r05-general-linear-map-0097

Original question and givens

Use the complete rectangular matrix A=[[9, -3, 3], [-1, -9, -5]] 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, -3], v=[5, 1, -1], alpha=-1, 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=[[9, -3, 3], [-1, -9, -5]] 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, -3], v=[5, 1, -1], alpha=-1, 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.

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(-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(-1, -15)

After

input combination coordinate0 sum = -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, 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(-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(1, -3)

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, -3)

After

input combination coordinate2 first 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, S006

S008 · calculation

Before

input combination coordinate2 second scaled := mul(-3, -1)

After

input combination coordinate2 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, S007

S009 · calculation

Before

input combination coordinate2 sum := sum(3, 3)

After

input combination coordinate2 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, S008

S010 · implies

Before

combined image:matrix=[[9, -3, 3], [-1, -9, -5]];input=[-16, -2, 6]

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(9, -16)

After

combined image row0 column0 term = -144

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, -2)

After

combined 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, S011

S013 · calculation

Before

combined image row0 column2 term := mul(3, 6)

After

combined image row0 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, S012

S014 · calculation

Before

combined image row0 sum := sum(-144, 6, 18)

After

combined image row0 sum = -120

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, -16)

After

combined image row1 column0 term = 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, S014

S016 · calculation

Before

combined image row1 column1 term := mul(-9, -2)

After

combined image row1 column1 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, S015

S017 · calculation

Before

combined image row1 column2 term := mul(-5, 6)

After

combined image row1 column2 term = -30

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(16, 18, -30)

After

combined image row1 sum = 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, S017

S019 · implies

Before

u image:matrix=[[9, -3, 3], [-1, -9, -5]];input=[1, -1, -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(9, 1)

After

u image row0 column0 term = 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, S019

S021 · calculation

Before

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

After

u 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, S020

S022 · calculation

Before

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

After

u image row0 column2 term = -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, S021

S023 · calculation

Before

u image row0 sum := sum(9, 3, -9)

After

u image row0 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, 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(-9, -1)

After

u image row1 column1 term = 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, S024

S026 · calculation

Before

u image row1 column2 term := mul(-5, -3)

After

u image row1 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, S025

S027 · calculation

Before

u image row1 sum := sum(-1, 9, 15)

After

u image row1 sum = 23

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=[[9, -3, 3], [-1, -9, -5]];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(9, 5)

After

v image row0 column0 term = 45

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(3, -1)

After

v image row0 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, S030

S032 · calculation

Before

v image row0 sum := sum(45, -3, -3)

After

v image row0 sum = 39

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(-9, 1)

After

v image row1 column1 term = -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, S033

S035 · calculation

Before

v image row1 column2 term := mul(-5, -1)

After

v image row1 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, S034

S036 · calculation

Before

v image row1 sum := sum(-5, -9, 5)

After

v image row1 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, S035

S037 · calculation

Before

output combination coordinate0 first scaled := mul(-1, 3)

After

output combination coordinate0 first 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, S036

S038 · calculation

Before

output combination coordinate0 second scaled := mul(-3, 39)

After

output combination coordinate0 second scaled = -117

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(-3, -117)

After

output combination coordinate0 sum = -120

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, 23)

After

output combination coordinate1 first scaled = -23

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, -9)

After

output combination coordinate1 second scaled = 27

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(-23, 27)

After

output combination coordinate1 sum = 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, S041

S043 · implies

Before

matrix formula;alpha=-1;beta=-3;lhs=[-120, 4];rhs=[-120, 4]

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, -3, [-16, -2, 6], [-120, 4], [-120, 4], 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: 4eea069621866cfac6fc5a1fb1f50baadaed3b50f288bc6855d3c2e90a380ca0
  • candidate record SHA-256: 068b4a57e2791c9b13a5d085d47489790497473d4d94ad25865fdbc3d6bf1603
  • semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5
  • question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON