# Question 18733: Every Dual-Basis Pairing Must Be a Kronecker Delta [Return to catalogue page 94](https://ocme.evemisslab.com/en/questions/page/94/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261105-r02-dual-vector-space-foundations/#q-20261014-advance-20261105-r02-dual-vector-space-0033) 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. R0036-18733 · q-20261014-advance-20261105-r02-dual-vector-space-0033 ## Original question and givens Use ordered basis column vectors=[[1, 0], [8, 1]] and candidate dual rows=[[1, -8], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate). - G001 — Use ordered basis column vectors=[[1, 0], [8, 1]] and candidate dual rows=[[1, -8], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate). ## 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-dual-vector-space](https://ocme.evemisslab.com/en/mko/mko-dual-vector-space/) ## Step-by-step derivation S001 · impliesBefore source operation=dual-dual-basis-certificate;original snapshot and mathematics retained After explicit ordered source output fields=[pairing_matrix, is_dual_basis, certificate] Conditions: none Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore dual original ordered basis:complete ordered matrix=[[1, 8], [0, 1]];rows=2;columns=2 After use exact invertible ROW operations only;original COLUMN roles/order stay fixed Conditions: none Keep original basis vectors in ordered columns.Semantic sorting must not reorder coefficients or candidate rows.Row operations multiply by an invertible map,retaining kernel,column relations and rank,not the literal original column space. Step identifiers and dependenciesRule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S001 S003 · calculationBefore dual original ordered basis pivot0 column0 normalized entry0 := div(1, 1) After dual original ordered basis pivot0 column0 normalized entry0 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore dual original ordered basis pivot0 column0 normalized entry1 := div(8, 1) After dual original ordered basis pivot0 column0 normalized entry1 = 8 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore dual original ordered basis:pivot row=0;column=0;nonzero pivot=1;old row=[1, 8] After normalized full row=[1, 8] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S004 S006 · calculationBefore dual original ordered basis pivot1 column1 normalized entry0 := div(0, 1) After dual original ordered basis pivot1 column1 normalized entry0 = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore dual original ordered basis pivot1 column1 normalized entry1 := div(1, 1) After dual original ordered basis pivot1 column1 normalized entry1 = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore dual original ordered basis:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1] After normalized full row=[0, 1] Conditions: none Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs. Step identifiers and dependenciesRule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S007 S009 · calculationBefore dual original ordered basis eliminate row0 pivot1 entry0 product := mul(8, 0) After dual original ordered basis eliminate row0 pivot1 entry0 product = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore dual original ordered basis eliminate row0 pivot1 entry0 difference := sub(1, 0) After dual original ordered basis eliminate row0 pivot1 entry0 difference = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore dual original ordered basis eliminate row0 pivot1 entry1 product := mul(8, 1) After dual original ordered basis eliminate row0 pivot1 entry1 product = 8 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore dual original ordered basis eliminate row0 pivot1 entry1 difference := sub(8, 8) After dual original ordered basis eliminate row0 pivot1 entry1 difference = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore dual original ordered basis:target row=0;old=[1, 8];factor=8;pivot row=[0, 1] After new full row=[1, 0] Conditions: none Subtract the same pivot-row multiple in every entry;adding it back is the inverse. Step identifiers and dependenciesRule: finite_linear_invertible_all_entry_elimination · Uses: G001, S012 S014 · impliesBefore dual original ordered basis:complete exact reduced matrix=[[1, 0], [0, 1]];pivot columns=[0, 1] After rank=2;pivot columns independent and all original column relations preserved;zero rows retained Conditions: none Distinct pivot identity coordinates prove independence;other columns are generated by their pivot-row coordinates.Invertible operations transport column relations back to the original matrix.Keep zero rows;do not infer rank from sampled vectors. Step identifiers and dependenciesRule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S013 S015 · impliesBefore dual original ordered basis:complete ORIGINAL independent columns=[[1, 0], [8, 1]] After declared subspace dimension=2;basis of its full real span;empty basis means zero subspace Conditions: none The source declares a full basis and exact rank checks independence,so its count is the subspace dimension.An empty basis spans the zero subspace;the finite list is not the set of all subspace elements. Step identifiers and dependenciesRule: directsum_declared_basis_complete_real_span · Uses: G001, S014 S016 · calculationBefore dual candidate row0 basis column0 coordinate0 product := mul(1, 1) After dual candidate row0 basis column0 coordinate0 product = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore dual candidate row0 basis column0 coordinate1 product := mul(-8, 0) After dual candidate row0 basis column0 coordinate1 product = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore dual candidate row0 basis column0 complete sum := sum(1, 0) After dual candidate row0 basis column0 complete sum = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore dual candidate row0 basis column1 coordinate0 product := mul(1, 8) After dual candidate row0 basis column1 coordinate0 product = 8 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore dual candidate row0 basis column1 coordinate1 product := mul(-8, 1) After dual candidate row0 basis column1 coordinate1 product = -8 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore dual candidate row0 basis column1 complete sum := sum(8, -8) After dual candidate row0 basis column1 complete sum = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore dual candidate row1 basis column0 coordinate0 product := mul(0, 1) After dual candidate row1 basis column0 coordinate0 product = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore dual candidate row1 basis column0 coordinate1 product := mul(1, 0) After dual candidate row1 basis column0 coordinate1 product = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore dual candidate row1 basis column0 complete sum := sum(0, 0) After dual candidate row1 basis column0 complete sum = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore dual candidate row1 basis column1 coordinate0 product := mul(0, 8) After dual candidate row1 basis column1 coordinate0 product = 0 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore dual candidate row1 basis column1 coordinate1 product := mul(1, 1) After dual candidate row1 basis column1 coordinate1 product = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore dual candidate row1 basis column1 complete sum := sum(0, 1) After dual candidate row1 basis column1 complete sum = 1 Conditions: none Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore ORIGINAL basis columns=[[1, 0], [8, 1]];candidate functional ROWS=[[1, -8], [0, 1]];ALL phi_i(v_j)=[[1, 0], [0, 1]] After complete Kronecker delta identity=[[1, 0], [0, 1]];candidate is dual basis=Yes;not just sampled pairings Conditions: none With a complete independent original basis,all delta pairings make each functional extract the corresponding coordinate of every basis expansion,hence the full dual basis.Conversely a dual basis must satisfy every pairing.False rejects this candidate,not existence of the actual dual basis.In dimension0 the identity matrix is empty. Step identifiers and dependenciesRule: dual_all_pairings_not_primal_identification_or_nonexistence · Uses: G001, S027 ## Reverse checks - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Recompute dual candidate row1 basis column1 complete sum with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute dual candidate row1 basis column1 coordinate1 product with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute dual candidate row1 basis column1 coordinate0 product with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute dual candidate row1 basis column0 complete sum with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute dual candidate row1 basis column0 coordinate1 product with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute dual candidate row1 basis column0 coordinate0 product with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute dual candidate row0 basis column1 complete sum with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute dual candidate row0 basis column1 coordinate1 product with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute dual candidate row0 basis column1 coordinate0 product with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute dual candidate row0 basis column0 complete sum with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute dual candidate row0 basis column0 coordinate1 product with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute dual candidate row0 basis column0 coordinate0 product with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute dual original ordered basis eliminate row0 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute dual original ordered basis eliminate row0 pivot1 entry1 product with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute dual original ordered basis eliminate row0 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute dual original ordered basis eliminate row0 pivot1 entry0 product with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute dual original ordered basis pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute dual original ordered basis pivot1 column1 normalized entry0 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 dual original ordered basis pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute dual original ordered basis pivot0 column0 normalized entry0 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, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta) 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: 465468dffb6d9704f15b54c32c90114c19badaa56bf9b2c6ce3eac3e4fb56410 - candidate record SHA-256: 108f5f5178b72ec5bbd4f64c50c06cb0de3a53c9a4d3bc705624a5c021db96fa - semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301 - question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r02-dual-vector-space-0033.json)