Question 10836: A Single Generator, Including the Zero-Vector Case

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. generator=[2, 4], target=[0, 0]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

• 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. generator=[2, 4], target=[0, 0]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.

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

single-generator first nonzero pivot scalar := div(0, 2)

single-generator first nonzero pivot scalar = 0

generator=[2, 4];target=[0, 0];first nonzero coordinate=0

necessary scalar=0

single-generator coordinate0 scaled := mul(0, 2)

single-generator coordinate0 scaled = 0

single-generator coordinate0 residual := sub(0, 0)

single-generator coordinate0 residual = 0

single-generator coordinate1 scaled := mul(0, 4)

single-generator coordinate1 scaled = 0

single-generator coordinate1 residual := sub(0, 0)

single-generator coordinate1 residual = 0

pivot scalar=0;all coordinate residuals=[0, 0]

in single-generator span=Yes;returned scalar remains=0

Reverse checks

• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute single-generator coordinate1 residual with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute single-generator coordinate1 scaled with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute single-generator coordinate0 residual with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute single-generator coordinate0 scaled with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute single-generator first nonzero pivot scalar 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.

Result

([2, 4], [0, 0], 0, 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.