# Question 10885: A Given Coefficient Witness against a Target Vector [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-0035) 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. R0020-10885 · q-20261013-advance-20261023-r02-linear-combination-span-0035 ## 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=[-3, -1, 2], target=[-1, 1]. 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=[-3, -1, 2], target=[-1, 1]. 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](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 · impliesBefore original ordered generators=[[1, 0], [0, 1], [1, 1]];aligned coefficients=[-3, -1, 2] After fixed coordinate dimension=2;generator duplicates/order retained Conditions: none Pair each coefficient with its own generator,retaining duplicates and order.Do not sort/deduplicate by semantic identity or replace a finite combination by an infinite series. Step identifiers and dependenciesRule: span_generator_coefficient_pairing · Uses: G001, S001 S003 · calculationBefore generator0 coordinate0 contribution := mul(-3, 1) After generator0 coordinate0 contribution = -3 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, S002 S004 · calculationBefore generator1 coordinate0 contribution := mul(-1, 0) After generator1 coordinate0 contribution = 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, S003 S005 · calculationBefore generator2 coordinate0 contribution := mul(2, 1) After generator2 coordinate0 contribution = 2 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 weighted coordinate0 complete sum := sum(-3, 0, 2) After weighted coordinate0 complete sum = -1 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 generator0 coordinate1 contribution := mul(-3, 0) After generator0 coordinate1 contribution = 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 · calculationBefore generator1 coordinate1 contribution := mul(-1, 1) After generator1 coordinate1 contribution = -1 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, S007 S009 · calculationBefore generator2 coordinate1 contribution := mul(2, 1) After generator2 coordinate1 contribution = 2 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, S008 S010 · calculationBefore weighted coordinate1 complete sum := sum(0, -1, 2) After weighted coordinate1 complete sum = 1 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, S009 S011 · impliesBefore all ordered weighted coordinates=[-1, 1] After finite combination=[-1, 1] Conditions: none Add every generator contribution at the same coordinate;an inner product or norm is not the requested vector. Step identifiers and dependenciesRule: span_complete_componentwise_reconstruction · Uses: G001, S010 S012 · calculationBefore given witness coordinate0 residual := sub(-1, -1) After given witness 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, S011 S013 · calculationBefore given witness coordinate1 residual := sub(1, 1) After given witness 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, S012 S014 · impliesBefore target=[-1, 1];given-witness reconstruction=[-1, 1];residuals=[0, 0] After this coefficient witness matches=Yes Conditions: none Every residual must be0 for this coefficient witness.A failure rejects this witness only,not membership in the span. Step identifiers and dependenciesRule: failed_given_span_witness_not_nonmembership · Uses: G001, S013 ## 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]], [-3, -1, 2], [-1, 1], [-1, 1], 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: 01978d1291ba36745562bd3206ced40c34f6b730970c4da11bac6143e61f825d - candidate record SHA-256: bb921fe37cb561a5098b698e6659d2618d4f121329c6fe3d4df56eb28a9e219f - 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-0035.json)