Question 11218: Independence and Spanning Are Separate Plane-Basis Obligations

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, 5], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.

• 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, 5], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.

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

ordered column vectors=[[1, 5], [0, 1]]

homogeneous coefficient matrix=[[1, 0], [5, 1]];unknown count=2

matrix=[[1, 0], [5, 1]];next pivot column=0;swap rows 0,0

matrix=[[1, 0], [5, 1]]

pivot0 normalized column0 := div(1, 1)

pivot0 normalized column0 = 1

pivot0 normalized column1 := div(0, 1)

pivot0 normalized column1 = 0

row=0;old row=[1, 0];nonzero divisor=1

new row=[1, 0]

eliminate pivot0 row1 column0 scaled := mul(5, 1)

eliminate pivot0 row1 column0 scaled = 5

eliminate pivot0 row1 column0 residual := sub(5, 5)

eliminate pivot0 row1 column0 residual = 0

eliminate pivot0 row1 column1 scaled := mul(5, 0)

eliminate pivot0 row1 column1 scaled = 0

eliminate pivot0 row1 column1 residual := sub(1, 0)

eliminate pivot0 row1 column1 residual = 1

target row=1;old=[5, 1];pivot row=[1, 0];factor=5

new row=[0, 1];matrix=[[1, 0], [0, 1]]

matrix=[[1, 0], [0, 1]];next pivot column=1;swap rows 1,1

matrix=[[1, 0], [0, 1]]

pivot1 normalized column0 := div(0, 1)

pivot1 normalized column0 = 0

pivot1 normalized column1 := div(1, 1)

pivot1 normalized column1 = 1

row=1;old row=[0, 1];nonzero divisor=1

new row=[0, 1]

reduced matrix=[[1, 0], [0, 1]];pivot columns=[0, 1];unknown count=2

rank=2;free variables=0;independent=Yes

ambient space=R2;rank=2;family size=2

independent=Yes;spans_R2=Yes;basis_R2=Yes

Reverse checks

• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute pivot1 normalized column1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute pivot1 normalized column0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 eliminate pivot0 row1 column1 residual with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute eliminate pivot0 row1 column1 scaled with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute eliminate pivot0 row1 column0 residual with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute eliminate pivot0 row1 column0 scaled 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 pivot0 normalized column1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute pivot0 normalized column0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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, 5], [0, 1]], 2, 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.