# Question 11237: Unique Coordinates Only after Qualifying an Ordered Plane Basis [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-0062) 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. R0021-11237 · q-20261013-advance-20261024-r01-linear-independence-basis-0062 ## 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. ordered basis=[[1, -1], [0, 1]], target=[-1, 1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. - 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. ordered basis=[[1, -1], [0, 1]], target=[-1, 1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. ## 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 · calculationBefore basis determinant first product := mul(1, 1) After basis determinant first product = 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 S002 · calculationBefore basis determinant second product := mul(0, -1) After basis determinant second product = 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, S001 S003 · calculationBefore basis determinant signed difference := sub(1, 0) After basis determinant signed difference = 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, S002 S004 · impliesBefore ordered basis columns=[[1, -1], [0, 1]];target=[-1, 1];signed determinant=1 After nonzero determinant;unique real coordinates;Cramer division permitted Conditions: none A nonzero signed determinant gives unique coordinates.Solve in the original basis order,without taking an absolute value or assuming invertibility. Step identifiers and dependenciesRule: ordered_basis_qualification_before_coordinates · Uses: G001, S003 S005 · calculationBefore first coordinate numerator first product := mul(-1, 1) After first coordinate numerator first product = -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, S004 S006 · calculationBefore first coordinate numerator second product := mul(0, 1) After first coordinate numerator second product = 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 first coordinate numerator signed difference := sub(-1, 0) After first coordinate numerator signed difference = -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, S006 S008 · calculationBefore second coordinate numerator first product := mul(1, 1) After second coordinate numerator first product = 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, S007 S009 · calculationBefore second coordinate numerator second product := mul(-1, -1) After second coordinate numerator second product = 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, S008 S010 · calculationBefore second coordinate numerator signed difference := sub(1, 1) After second coordinate numerator signed difference = 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, S009 S011 · calculationBefore first basis coordinate := div(-1, 1) After first basis coordinate = -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, S010 S012 · calculationBefore second basis coordinate := div(0, 1) After second basis coordinate = 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, S011 S013 · impliesBefore coordinate reconstruction:ordered vectors=[[1, -1], [0, 1]];aligned weights=[-1, 0] 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, S012 S014 · calculationBefore coordinate reconstruction coordinate0 term0 := mul(-1, 1) After coordinate reconstruction 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, S013 S015 · calculationBefore coordinate reconstruction coordinate0 term1 := mul(0, 0) After coordinate reconstruction 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, S014 S016 · calculationBefore coordinate reconstruction coordinate0 sum := sum(-1, 0) After coordinate reconstruction coordinate0 sum = -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, S015 S017 · calculationBefore coordinate reconstruction coordinate1 term0 := mul(-1, -1) After coordinate reconstruction coordinate1 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, S016 S018 · calculationBefore coordinate reconstruction coordinate1 term1 := mul(0, 1) After coordinate reconstruction coordinate1 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, S017 S019 · calculationBefore coordinate reconstruction coordinate1 sum := sum(1, 0) After coordinate reconstruction coordinate1 sum = 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, S018 S020 · calculationBefore basis target residual0 := sub(-1, -1) After basis target residual0 = 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, S019 S021 · calculationBefore basis target residual1 := sub(1, 1) After basis target residual1 = 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, S020 S022 · impliesBefore coordinates=[-1, 0];reconstructed=[-1, 1];target=[-1, 1];residuals=[0, 0] After both original equations satisfied;uniqueness follows from qualified basis Conditions: none Reconstructing both original equations proves existence.Uniqueness follows separately from the nonzero determinant,not from one successful substitution. Step identifiers and dependenciesRule: coordinate_reconstruction_and_uniqueness · Uses: G001, S021 ## Reverse checks - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Recompute basis target residual1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute basis target residual0 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute coordinate reconstruction coordinate1 sum with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute coordinate reconstruction coordinate1 term1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute coordinate reconstruction coordinate1 term0 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute coordinate reconstruction coordinate0 sum with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute coordinate reconstruction coordinate0 term1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute coordinate reconstruction coordinate0 term0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute second basis coordinate with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute first basis coordinate with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute second coordinate numerator signed difference with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute second coordinate numerator second product with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute second coordinate numerator first product with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute first coordinate numerator signed difference with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute first coordinate numerator second product with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute first coordinate numerator first product with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Recompute basis determinant signed difference with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute basis determinant second product with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute basis determinant first product with exact rational arithmetic, not rounded decimals. ## Result ([[1, -1], [0, 1]], [-1, 1], 1, -1, 0, [-1, 1]) 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: 7e8b2b337dbd5a5a6b700e279843236d36024c787256c7e705d1cfccfedbd9aa - candidate record SHA-256: b84a8d9421fde508135d5ec32ea70839773d5b5abed6d96fb1b81ccb9f0364b3 - 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-0062.json)