# Question 11201: Independence and Spanning Are Separate Plane-Basis Obligations [Return to catalogue page 57](https://ocme.evemisslab.com/en/questions/page/57/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations/#q-20261013-advance-20261024-r01-linear-independence-basis-0026) 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. R0021-11201 · q-20261013-advance-20261024-r01-linear-independence-basis-0026 ## 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, -12], [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, -12], [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](https://ocme.evemisslab.com/en/mko/mko-linear-independence-basis/) ## Step-by-step derivation S001 · impliesBefore ordered column vectors=[[1, -12], [0, 1]] After homogeneous coefficient matrix=[[1, 0], [-12, 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. Step identifiers and dependenciesRule: column_matrix_homogeneous_equations · Uses: G001 S002 · impliesBefore matrix=[[1, 0], [-12, 1]];next pivot column=0;swap rows 0,0 After matrix=[[1, 0], [-12, 1]] Conditions: none Swapping rows only reorders equations;the same swap reverses it.It does not exchange the original vector columns. Step identifiers and dependenciesRule: invertible_row_swap · Uses: G001, S001 S003 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore 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. Step identifiers and dependenciesRule: nonzero_row_scaling_solution_equivalence · Uses: G001, S004 S006 · calculationBefore eliminate pivot0 row1 column0 scaled := mul(-12, 1) After eliminate pivot0 row1 column0 scaled = -12 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore eliminate pivot0 row1 column0 residual := sub(-12, -12) After eliminate pivot0 row1 column0 residual = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore eliminate pivot0 row1 column1 scaled := mul(-12, 0) After eliminate pivot0 row1 column1 scaled = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore eliminate pivot0 row1 column1 residual := sub(1, 0) After eliminate pivot0 row1 column1 residual = 1 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore target row=1;old=[-12, 1];pivot row=[1, 0];factor=-12 After new row=[0, 1];matrix=[[1, 0], [0, 1]] Conditions: none Subtract factor times the pivot row to eliminate this entry.Adding it back recovers the original equation;no solutions are created or lost. Step identifiers and dependenciesRule: invertible_pivot_row_subtraction · Uses: G001, S009 S011 · impliesBefore 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. Step identifiers and dependenciesRule: invertible_row_swap · Uses: G001, S010 S012 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore 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. Step identifiers and dependenciesRule: nonzero_row_scaling_solution_equivalence · Uses: G001, S013 S015 · impliesBefore 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. Step identifiers and dependenciesRule: rank_free_variables_and_independence · Uses: G001, S014 S016 · impliesBefore 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. Step identifiers and dependenciesRule: basis_independence_and_spanning_separate · Uses: G001, S015 ## 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, -12], [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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: a591afb038d98ad9a831ab4695b327938b9a0e67cc8420dcb16900bfb180504e - candidate record SHA-256: 34af50e536217dc1950742e8ee613999cdff78b546b7c1597e6b051cf39925d0 - semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258 - question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0026.json)