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R0014-07542 · q-20261012-advance-20261016-r03-linear-transformation-0092
Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[4, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).
G001 — Use the explicitly declared matrix model on all R²→R² with column vectors. Finite test values do not prove an arbitrary function linear; the zero matrix defines a linear map. Declare F(v)=Mv+b on all R², with M=[[1, 0], [0, 2]], v=[4, 1], b=[0, 0]. Give (Mv, Mv+b, F(0)=b, is this declared affine model linear?).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
M=[[1, 0], [0, 2]];v=[4, 1]
After
declared R2 matrix model;finite values are not a test of an arbitrary function
Conditions: none
The declared matrix model Mv gives a linear map on all R² by distributivity.The listed finite evaluations do not prove an arbitrary function linear.The zero map is linear;nonzero values or invertibility are not required.
Rule: declared_linear_model_scope · Uses: G001
Before
Mv[0] term0 := mul(1, 4)
After
Mv[0] term0 = 4
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Mv[0] term1 := mul(0, 1)
After
Mv[0] term1 = 0
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Mv[0] row sum := sum(4, 0)
After
Mv[0] row sum = 4
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Mv[1] term0 := mul(0, 4)
After
Mv[1] term0 = 0
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Mv[1] term1 := mul(2, 1)
After
Mv[1] term1 = 2
Conditions: none
Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
Mv[1] row sum := sum(0, 2)
After
Mv[1] row sum = 2
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Mv+b[0] := sum(4, 0)
After
Mv+b[0] = 4
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
Mv+b[1] := sum(2, 0)
After
Mv+b[1] = 2
Conditions: none
Add the already aligned component or position,without mixing coordinates.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
F(x)=Mx+b;M=[[1, 0], [0, 2]];b=[0, 0]
After
F(0)=[0, 0];linear model qualified=Yes
Conditions: none
Necessity:a linear map has F(0)=0,so b must be zero.Sufficiency:when b=0 the declared model reduces to Mx,whose additivity and homogeneity follow from distributivity.This is not a claim that a zero-value test proves an arbitrary function linear.
Rule: affine_zero_obstruction_and_sufficiency · Uses: G001, S009
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 Mv+b[1] with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute Mv+b[0] with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Mv[1] row sum with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Mv[1] term1 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Mv[1] term0 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Mv[0] row sum with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Mv[0] term1 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Mv[0] term0 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([4, 2], [4, 2], [0, 0], Yes)
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: 0e71dcc4644bc8490543b3acd010475a45e8837ef86d351fb1ba33c7a1dfa42dcandidate record SHA-256: 103c98c64e9e08191f720b1c6d541c59e1ddc76377da1f70b8aa586cbef2d500semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705Lean: NOT_RUN · human review: NOT_PERFORMED