Explore related knowledge and questions →
← Question derivation index · Original question batch
R0025-13071 · q-20261014-advance-20261027-r05-general-linear-map-0096
Use the complete rectangular matrix A=[[8, 3, 0], [-2, -8, 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, -2, 3], v=[4, 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=[[8, 3, 0], [-2, -8, 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, -2, 3], v=[4, 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).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.
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.
Rule: exact_rational_arithmetic · Uses: G001
Before
input combination coordinate0 second scaled := mul(3, 4)
After
input combination coordinate0 second scaled = 12
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
input combination coordinate0 sum := sum(-2, 12)
After
input combination coordinate0 sum = 10
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S002
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
input combination coordinate1 sum := sum(4, 3)
After
input combination coordinate1 sum = 7
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S005
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.
Rule: exact_rational_arithmetic · Uses: G001, S006
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.
Rule: exact_rational_arithmetic · Uses: G001, S007
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.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
combined image:matrix=[[8, 3, 0], [-2, -8, 3]];input=[10, 7, -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.
Rule: rectangular_linear_map_source_target_roles · Uses: G001, S009
Before
combined image row0 column0 term := mul(8, 10)
After
combined image row0 column0 term = 80
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
combined image row0 column1 term := mul(3, 7)
After
combined image row0 column1 term = 21
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
combined image row0 column2 term := mul(0, -12)
After
combined image row0 column2 term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
combined image row0 sum := sum(80, 21, 0)
After
combined image row0 sum = 101
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
combined image row1 column0 term := mul(-2, 10)
After
combined image row1 column0 term = -20
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
combined image row1 column1 term := mul(-8, 7)
After
combined image row1 column1 term = -56
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
combined image row1 column2 term := mul(3, -12)
After
combined image row1 column2 term = -36
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
combined image row1 sum := sum(-20, -56, -36)
After
combined image row1 sum = -112
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
u image:matrix=[[8, 3, 0], [-2, -8, 3]];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.
Rule: rectangular_linear_map_source_target_roles · Uses: G001, S018
Before
u image row0 column0 term := mul(8, 1)
After
u image row0 column0 term = 8
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S019
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.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
u image row0 column2 term := mul(0, 3)
After
u image row0 column2 term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
u image row0 sum := sum(8, -6, 0)
After
u image row0 sum = 2
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S022
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.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
u image row1 column1 term := mul(-8, -2)
After
u image row1 column1 term = 16
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
u image row1 column2 term := mul(3, 3)
After
u image row1 column2 term = 9
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
u image row1 sum := sum(-2, 16, 9)
After
u image row1 sum = 23
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
v image:matrix=[[8, 3, 0], [-2, -8, 3]];input=[4, 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.
Rule: rectangular_linear_map_source_target_roles · Uses: G001, S027
Before
v image row0 column0 term := mul(8, 4)
After
v image row0 column0 term = 32
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S028
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.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
v image row0 column2 term := mul(0, -2)
After
v image row0 column2 term = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
v image row0 sum := sum(32, 3, 0)
After
v image row0 sum = 35
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
v image row1 column0 term := mul(-2, 4)
After
v image row1 column0 term = -8
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
v image row1 column1 term := mul(-8, 1)
After
v image row1 column1 term = -8
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
v image row1 column2 term := mul(3, -2)
After
v image row1 column2 term = -6
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
v image row1 sum := sum(-8, -8, -6)
After
v image row1 sum = -22
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
output combination coordinate0 first scaled := mul(-2, 2)
After
output combination coordinate0 first scaled = -4
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
output combination coordinate0 second scaled := mul(3, 35)
After
output combination coordinate0 second scaled = 105
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
output combination coordinate0 sum := sum(-4, 105)
After
output combination coordinate0 sum = 101
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
output combination coordinate1 first scaled := mul(-2, 23)
After
output combination coordinate1 first scaled = -46
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
output combination coordinate1 second scaled := mul(3, -22)
After
output combination coordinate1 second scaled = -66
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
output combination coordinate1 sum := sum(-46, -66)
After
output combination coordinate1 sum = -112
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
matrix formula;alpha=-2;beta=3;lhs=[101, -112];rhs=[101, -112]
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.
Rule: matrix_combination_universal_coordinate_certificate · Uses: G001, S042
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.(-2, 3, [10, 7, -12], [101, -112], [101, -112], Yes, matrix_formula_universal_certificate)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 7b8e2aa7820189e99f974fc6e7db9ec77353a06681e2cf749972f2b2f7f4a638candidate record SHA-256: 92c852c8b1afe7728d1f11aca397ca736ab4e0ad0ec368e1d53c7f0ea6db00fbsemantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8aLean: NOT_RUN · human review: NOT_PERFORMED