# Question 10827: A Single Generator, Including the Zero-Vector Case [Return to catalogue page 55](https://ocme.evemisslab.com/en/questions/page/55/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261023-r02-linear-combination-span-foundations/#q-20261013-advance-20261023-r02-linear-combination-span-0077) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0020-10827 · q-20261013-advance-20261023-r02-linear-combination-span-0077 ## 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=[-18, -36]. 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=[-18, -36]. 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](https://ocme.evemisslab.com/en/mko/mko-linear-combination-span/) ## Step-by-step derivation S001 · impliesBefore fixed real coordinate space;span uses all finite real linear combinations After given coefficient witness and existence of some coefficients are separate Conditions: none Span quantifies all finite real linear combinations.These integer/rational instances are declared slices;a failed designated witness does not show every coefficient choice fails. Step identifiers and dependenciesRule: span_quantifier_and_witness_scope · Uses: G001 S002 · calculationBefore single-generator first nonzero pivot scalar := div(-18, 2) After single-generator first nonzero pivot scalar = -9 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore generator=[2, 4];target=[-18, -36];first nonzero coordinate=0 After necessary scalar=-9 Conditions: none The first nonzero pivot equation forces every possible scalar to equal this ratio,with a negative denominator normalized exactly. Step identifiers and dependenciesRule: single_generator_pivot_necessary_scalar · Uses: G001, S002 S004 · calculationBefore single-generator coordinate0 scaled := mul(-9, 2) After single-generator coordinate0 scaled = -18 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore single-generator coordinate0 residual := sub(-18, -18) After single-generator coordinate0 residual = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore single-generator coordinate1 scaled := mul(-9, 4) After single-generator coordinate1 scaled = -36 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore single-generator coordinate1 residual := sub(-36, -36) After single-generator coordinate1 residual = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore pivot scalar=-9;all coordinate residuals=[0, 0] After in single-generator span=Yes;returned scalar remains=-9 Conditions: none Both directions matter:all equalities give a witness;if any fails,the pivot already forces the unique scalar,so no other scalar repairs it.Retain the candidate ratio even on failure as specified. Step identifiers and dependenciesRule: single_generator_all_coordinates_after_pivot · Uses: G001, S007 ## 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], [-18, -36], -9, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 0aa2c8d1d10d28ee866eac38d6f00e9e2d50e83d7d1305b7e19e59097f33fa12 - candidate record SHA-256: 57befeb0cd0338fe299f38671add4a146e8c35b95d71c56ba3f7897cdca54972 - semantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f - question-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261023-r02-linear-combination-span-0077.json)