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R0021-11213 · q-20261013-advance-20261024-r01-linear-independence-basis-0038
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, 0], [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, 0], [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.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.
Before
ordered column vectors=[[1, 0], [0, 1]]
After
homogeneous coefficient matrix=[[1, 0], [0, 1]];unknown count=2
Conditions: none
Use vectors as columns in original order.Independence means this homogeneous system has only the zero coefficient solution;row operations must preserve its solution set.
Rule: column_matrix_homogeneous_equations · Uses: G001
Before
matrix=[[1, 0], [0, 1]];next pivot column=0;swap rows 0,0
After
matrix=[[1, 0], [0, 1]]
Conditions: none
Swapping rows only reorders equations;the same swap reverses it.It does not exchange the original vector columns.
Rule: invertible_row_swap · Uses: G001, S001
Before
pivot0 normalized column0 := div(1, 1)
After
pivot0 normalized column0 = 1
Conditions: none
Compute exactly with this step's specified roles,retaining signs,denominators and original pairings.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
pivot0 normalized column1 := div(0, 1)
After
pivot0 normalized column1 = 0
Conditions: none
Compute exactly with this step's specified roles,retaining signs,denominators and original pairings.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
row=0;old row=[1, 0];nonzero divisor=1
After
new row=[1, 0]
Conditions: none
The pivot is known nonzero before division of the whole row.Multiplication by it reverses the operation,so the homogeneous solution set is unchanged.
Rule: nonzero_row_scaling_solution_equivalence · Uses: G001, S004
Before
matrix=[[1, 0], [0, 1]];next pivot column=1;swap rows 1,1
After
matrix=[[1, 0], [0, 1]]
Conditions: none
Swapping rows only reorders equations;the same swap reverses it.It does not exchange the original vector columns.
Rule: invertible_row_swap · Uses: G001, S005
Before
pivot1 normalized column0 := div(0, 1)
After
pivot1 normalized column0 = 0
Conditions: none
Compute exactly with this step's specified roles,retaining signs,denominators and original pairings.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
pivot1 normalized column1 := div(1, 1)
After
pivot1 normalized column1 = 1
Conditions: none
Compute exactly with this step's specified roles,retaining signs,denominators and original pairings.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
row=1;old row=[0, 1];nonzero divisor=1
After
new row=[0, 1]
Conditions: none
The pivot is known nonzero before division of the whole row.Multiplication by it reverses the operation,so the homogeneous solution set is unchanged.
Rule: nonzero_row_scaling_solution_equivalence · Uses: G001, S008
Before
reduced matrix=[[1, 0], [0, 1]];pivot columns=[0, 1];unknown count=2
After
rank=2;free variables=0;independent=Yes
Conditions: none
Rank is the pivot count.If every column pivots,the homogeneous solution is zero only.Otherwise set one free variable to1,the other free variables to0,and back-substitute pivot variables to obtain a nontrivial zero combination.
Rule: rank_free_variables_and_independence · Uses: G001, S009
Before
ambient space=R2;rank=2;family size=2
After
independent=Yes;spans_R2=Yes;basis_R2=Yes
Conditions: none
Invertible row operations preserve image dimension;rank2 is required to span R².An independent singleton does not span the plane,and redundant spanning vectors are not a basis.
Rule: basis_independence_and_spanning_separate · Uses: G001, S010
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute pivot1 normalized column1 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute pivot1 normalized column0 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([[1, 0], [0, 1]], 2, Yes, Yes, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: cc189fa6b9a8a24ca29f7633d64a6308bf09e2f0c19fea13e9d971334e84b70bcandidate record SHA-256: c4ba72c5fcdb800f762c31a4a2ec8f2bc9509af4a1047011130ffd6dc199b5ddsemantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139Lean: NOT_RUN · human review: NOT_PERFORMED