Question 10862: An Ordered Generator-Coefficient Weighted Sum

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=[-1, 1, -1]. Give (original vectors,original coefficients,the coordinate sum).

• 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=[-1, 1, -1]. Give (original vectors,original coefficients,the coordinate sum).

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=[-1, 1, -1]

fixed coordinate dimension=2;generator duplicates/order retained

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

generator0 coordinate0 contribution = -1

generator1 coordinate0 contribution := mul(1, 0)

generator1 coordinate0 contribution = 0

generator2 coordinate0 contribution := mul(-1, 1)

generator2 coordinate0 contribution = -1

weighted coordinate0 complete sum := sum(-1, 0, -1)

weighted coordinate0 complete sum = -2

generator0 coordinate1 contribution := mul(-1, 0)

generator0 coordinate1 contribution = 0

generator1 coordinate1 contribution := mul(1, 1)

generator1 coordinate1 contribution = 1

generator2 coordinate1 contribution := mul(-1, 1)

generator2 coordinate1 contribution = -1

weighted coordinate1 complete sum := sum(0, 1, -1)

weighted coordinate1 complete sum = 0

all ordered weighted coordinates=[-2, 0]

finite combination=[-2, 0]

Reverse checks

• 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]], [-1, 1, -1], [-2, 0])

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.