Question 10814: Unique Real Coordinates in Two Independent Plane Generators

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=[1, 3]. 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=[1, 3]. 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

Step-by-step derivation

fixed real coordinate space;span uses all finite real linear combinations

given coefficient witness and existence of some coefficients are separate

column determinant first product := mul(1, -1)

column determinant first product = -1

column determinant second product := mul(1, 1)

column determinant second product = 1

signed column determinant := sub(-1, 1)

signed column determinant = -2

first column=[1, 1];second column=[1, -1];target=[1, 3]

signed determinant=-2;independent generators=Yes

first coordinate numerator first product := mul(1, -1)

first coordinate numerator first product = -1

first coordinate numerator second product := mul(1, 3)

first coordinate numerator second product = 3

first coordinate numerator := sub(-1, 3)

first coordinate numerator = -4

second coordinate numerator first product := mul(1, 3)

second coordinate numerator first product = 3

second coordinate numerator second product := mul(1, 1)

second coordinate numerator second product = 1

second coordinate numerator := sub(3, 1)

second coordinate numerator = 2

coefficient of first generator := div(-4, -2)

coefficient of first generator = 2

coefficient of second generator := div(2, -2)

coefficient of second generator = -1

R2 reconstructed coordinate0 first contribution := mul(2, 1)

R2 reconstructed coordinate0 first contribution = 2

R2 reconstructed coordinate0 second contribution := mul(-1, 1)

R2 reconstructed coordinate0 second contribution = -1

R2 reconstructed coordinate0 sum := sum(2, -1)

R2 reconstructed coordinate0 sum = 1

R2 reconstructed coordinate1 first contribution := mul(2, 1)

R2 reconstructed coordinate1 first contribution = 2

R2 reconstructed coordinate1 second contribution := mul(-1, -1)

R2 reconstructed coordinate1 second contribution = 1

R2 reconstructed coordinate1 sum := sum(2, 1)

R2 reconstructed coordinate1 sum = 3

first-generator coefficient=2;second-generator coefficient=-1

reconstructed target=[1, 3]

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], [1, 3], -2, 2, -1, [1, 3])

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.