# Question 10806: Unique Real Coordinates in Two Independent Plane Generators [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-0056) 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. R0020-10806 · q-20261013-advance-20261023-r02-linear-combination-span-0056 ## 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. v1=[1, 1], v2=[1, -1], target=[-7, 2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). 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. v1=[1, 1], v2=[1, -1], target=[-7, 2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). 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 column determinant first product := mul(1, -1) After column determinant first product = -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, S001 S003 · calculationBefore column determinant second product := mul(1, 1) After column determinant second product = 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, S002 S004 · calculationBefore signed column determinant := sub(-1, 1) After signed column determinant = -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, S003 S005 · impliesBefore first column=[1, 1];second column=[1, -1];target=[-7, 2] After signed determinant=-2;independent generators=Yes Conditions: none Use the generators as columns in their original order.A nonzero signed determinant permits unique coordinates;do not take its absolute value or exchange roles. Step identifiers and dependenciesRule: R2_column_independence_before_division · Uses: G001, S004 S006 · calculationBefore first coordinate numerator first product := mul(-7, -1) After first coordinate numerator first product = 7 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 first coordinate numerator second product := mul(1, 2) After first coordinate numerator second product = 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, S006 S008 · calculationBefore first coordinate numerator := sub(7, 2) After first coordinate numerator = 5 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 second coordinate numerator first product := mul(1, 2) After second coordinate numerator first product = 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 second coordinate numerator second product := mul(-7, 1) After second coordinate numerator second product = -7 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 · calculationBefore second coordinate numerator := sub(2, -7) After second coordinate numerator = 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, S010 S012 · calculationBefore coefficient of first generator := div(5, -2) After coefficient of first generator = -5/2 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, S011 S013 · calculationBefore coefficient of second generator := div(9, -2) After coefficient of second generator = -9/2 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, S012 S014 · calculationBefore R2 reconstructed coordinate0 first contribution := mul(-5/2, 1) After R2 reconstructed coordinate0 first contribution = -5/2 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, S013 S015 · calculationBefore R2 reconstructed coordinate0 second contribution := mul(-9/2, 1) After R2 reconstructed coordinate0 second contribution = -9/2 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, S014 S016 · calculationBefore R2 reconstructed coordinate0 sum := sum(-5/2, -9/2) After R2 reconstructed coordinate0 sum = -7 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, S015 S017 · calculationBefore R2 reconstructed coordinate1 first contribution := mul(-5/2, 1) After R2 reconstructed coordinate1 first contribution = -5/2 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, S016 S018 · calculationBefore R2 reconstructed coordinate1 second contribution := mul(-9/2, -1) After R2 reconstructed coordinate1 second contribution = 9/2 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, S017 S019 · calculationBefore R2 reconstructed coordinate1 sum := sum(-5/2, 9/2) After R2 reconstructed coordinate1 sum = 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, S018 S020 · impliesBefore first-generator coefficient=-5/2;second-generator coefficient=-9/2 After reconstructed target=[-7, 2] Conditions: none Substitute both coefficients back into the original columns and recover both target coordinates.Denominator normalization does not change generator order. Step identifiers and dependenciesRule: R2_exact_coordinate_back_substitution · Uses: G001, S019 ## Reverse checks - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute R2 reconstructed coordinate1 sum with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute R2 reconstructed coordinate1 second contribution with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute R2 reconstructed coordinate1 first contribution with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute R2 reconstructed coordinate0 sum with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute R2 reconstructed coordinate0 second contribution with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute R2 reconstructed coordinate0 first contribution with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute coefficient of second generator with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute coefficient of first generator with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute second coordinate numerator with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute second coordinate numerator second product with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute second coordinate numerator first product with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute first coordinate numerator with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute first coordinate numerator second product with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute first coordinate numerator first product with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute signed column determinant with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute column determinant second product with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute column determinant first product 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, 1], [1, -1], [-7, 2], -2, -5/2, -9/2, [-7, 2]) 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: 3f2c0f0fb621e850c87204c9aa16b4e3e113297cb62e218199bc0445f2fd69cd - candidate record SHA-256: 50661110c8c00c74d0eb67a538f5c0ebba27c0aa8c2c155140135392ccf9df77 - 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-0056.json)