# Question 11270: A Zero Combination Must Also Be Nontrivial to Witness Dependence [Return to catalogue page 57](https://ocme.evemisslab.com/en/questions/page/57/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations/#q-20261013-advance-20261024-r01-linear-independence-basis-0095) 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. R0021-11270 · q-20261013-advance-20261024-r01-linear-independence-basis-0095 ## Original question and givens Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 7]], coefficients=[-1, -7, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?). - G001 — Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 7]], coefficients=[-1, -7, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?). ## 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-independence-basis](https://ocme.evemisslab.com/en/mko/mko-linear-independence-basis/) ## Step-by-step derivation S001 · impliesBefore dependence witness:ordered vectors=[[1, 0], [0, 1], [1, 7]];aligned weights=[-1, -7, 1] After dimension=2;preserve duplicates and order Conditions: none Preserve vector order,duplicates and coefficient pairing.Sum every vector's contribution within the same coordinate. Step identifiers and dependenciesRule: ordered_vector_weight_pairing · Uses: G001 S002 · calculationBefore dependence witness coordinate0 term0 := mul(-1, 1) After dependence witness coordinate0 term0 = -1 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore dependence witness coordinate0 term1 := mul(-7, 0) After dependence witness coordinate0 term1 = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore dependence witness coordinate0 term2 := mul(1, 1) After dependence witness coordinate0 term2 = 1 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore dependence witness coordinate0 sum := sum(-1, 0, 1) After dependence witness coordinate0 sum = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore dependence witness coordinate1 term0 := mul(-1, 0) After dependence witness coordinate1 term0 = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore dependence witness coordinate1 term1 := mul(-7, 1) After dependence witness coordinate1 term1 = -7 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore dependence witness coordinate1 term2 := mul(1, 7) After dependence witness coordinate1 term2 = 7 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore dependence witness coordinate1 sum := sum(0, -7, 7) After dependence witness coordinate1 sum = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore given coefficients=[-1, -7, 1];weighted sum=[0, 0] After nontrivial=Yes;zero_sum=Yes;dependence_witness=Yes Conditions: none A dependence witness needs both some nonzero coefficient and a zero vector sum.All-zero coefficients give only the trivial solution.A failed designated witness does not prove independence. Step identifiers and dependenciesRule: dependence_requires_nontrivial_zero_combination · Uses: G001, S009 ## Reverse checks - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute dependence witness coordinate1 sum with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute dependence witness coordinate1 term2 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute dependence witness coordinate1 term1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute dependence witness coordinate1 term0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute dependence witness coordinate0 sum with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute dependence witness coordinate0 term2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute dependence witness coordinate0 term1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute dependence witness coordinate0 term0 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 ([[1, 0], [0, 1], [1, 7]], [-1, -7, 1], [0, 0], Yes, Yes, 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: ad5fe5610a6b85ed1281e9fccb4846de7a786ad3ca2aa37b86c07f2de10e6106 - candidate record SHA-256: 89586b505c3fa00912202be194aa4d09b99487dcaab201dd381332220814e73c - semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258 - question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0095.json)