Question 10883: A Given Coefficient Witness against a Target Vector

Original question and givens

Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-5, -3, 0], target=[-5, -3]. Give (vectors,coefficients,target,reconstructed sum,matches?).

• G001 — Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-5, -3, 0], target=[-5, -3]. Give (vectors,coefficients,target,reconstructed sum,matches?).

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-linear-combination-span

Step-by-step derivation

fixed real coordinate space;span uses all finite real linear combinations

given coefficient witness and existence of some coefficients are separate

original ordered generators=[[1, 0], [0, 1], [1, 1]];aligned coefficients=[-5, -3, 0]

fixed coordinate dimension=2;generator duplicates/order retained

generator0 coordinate0 contribution := mul(-5, 1)

generator0 coordinate0 contribution = -5

generator1 coordinate0 contribution := mul(-3, 0)

generator1 coordinate0 contribution = 0

generator2 coordinate0 contribution := mul(0, 1)

generator2 coordinate0 contribution = 0

weighted coordinate0 complete sum := sum(-5, 0, 0)

weighted coordinate0 complete sum = -5

generator0 coordinate1 contribution := mul(-5, 0)

generator0 coordinate1 contribution = 0

generator1 coordinate1 contribution := mul(-3, 1)

generator1 coordinate1 contribution = -3

generator2 coordinate1 contribution := mul(0, 1)

generator2 coordinate1 contribution = 0

weighted coordinate1 complete sum := sum(0, -3, 0)

weighted coordinate1 complete sum = -3

all ordered weighted coordinates=[-5, -3]

finite combination=[-5, -3]

given witness coordinate0 residual := sub(-5, -5)

given witness coordinate0 residual = 0

given witness coordinate1 residual := sub(-3, -3)

given witness coordinate1 residual = 0

target=[-5, -3];given-witness reconstruction=[-5, -3];residuals=[0, 0]

this coefficient witness matches=Yes

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute given witness coordinate1 residual with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute given witness coordinate0 residual with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute weighted coordinate1 complete sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute generator2 coordinate1 contribution with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute generator1 coordinate1 contribution with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute generator0 coordinate1 contribution with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute weighted coordinate0 complete sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute generator2 coordinate0 contribution with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute generator1 coordinate0 contribution with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute generator0 coordinate0 contribution with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3], [-5, -3], Yes)

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.