Question 11234: Unique Coordinates Only after Qualifying an Ordered Plane Basis

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=[-4, -2]. 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=[-4, -2]. 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

Step-by-step derivation

basis determinant first product := mul(1, 1)

basis determinant first product = 1

basis determinant second product := mul(0, 1)

basis determinant second product = 0

basis determinant signed difference := sub(1, 0)

basis determinant signed difference = 1

ordered basis columns=[[1, 1], [0, 1]];target=[-4, -2];signed determinant=1

nonzero determinant;unique real coordinates;Cramer division permitted

first coordinate numerator first product := mul(-4, 1)

first coordinate numerator first product = -4

first coordinate numerator second product := mul(0, -2)

first coordinate numerator second product = 0

first coordinate numerator signed difference := sub(-4, 0)

first coordinate numerator signed difference = -4

second coordinate numerator first product := mul(1, -2)

second coordinate numerator first product = -2

second coordinate numerator second product := mul(-4, 1)

second coordinate numerator second product = -4

second coordinate numerator signed difference := sub(-2, -4)

second coordinate numerator signed difference = 2

first basis coordinate := div(-4, 1)

first basis coordinate = -4

second basis coordinate := div(2, 1)

second basis coordinate = 2

coordinate reconstruction:ordered vectors=[[1, 1], [0, 1]];aligned weights=[-4, 2]

dimension=2;preserve duplicates and order

coordinate reconstruction coordinate0 term0 := mul(-4, 1)

coordinate reconstruction coordinate0 term0 = -4

coordinate reconstruction coordinate0 term1 := mul(2, 0)

coordinate reconstruction coordinate0 term1 = 0

coordinate reconstruction coordinate0 sum := sum(-4, 0)

coordinate reconstruction coordinate0 sum = -4

coordinate reconstruction coordinate1 term0 := mul(-4, 1)

coordinate reconstruction coordinate1 term0 = -4

coordinate reconstruction coordinate1 term1 := mul(2, 1)

coordinate reconstruction coordinate1 term1 = 2

coordinate reconstruction coordinate1 sum := sum(-4, 2)

coordinate reconstruction coordinate1 sum = -2

basis target residual0 := sub(-4, -4)

basis target residual0 = 0

basis target residual1 := sub(-2, -2)

basis target residual1 = 0

coordinates=[-4, 2];reconstructed=[-4, -2];target=[-4, -2];residuals=[0, 0]

both original equations satisfied;uniqueness follows from qualified basis

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]], [-4, -2], 1, -4, 2, [-4, -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.