# Question 18713: Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws [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-0088) 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. R0036-18713 · q-20261014-advance-20261105-r02-dual-vector-space-0088 ## Original question and givens Use invertible change matrix P=[[1, 13], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[14, 3] and old covector row=[2, 13]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary). - G001 — Use invertible change matrix P=[[1, 13], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[14, 3] and old covector row=[2, 13]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary). ## 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-coordinate-change-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary] 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 change augmented P with identity:complete ordered matrix=[[1, 13, 1, 0], [0, 1, 0, 1]];rows=2;columns=4 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 change augmented P with identity pivot0 column0 normalized entry0 := div(1, 1) After dual change augmented P with identity 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 change augmented P with identity pivot0 column0 normalized entry1 := div(13, 1) After dual change augmented P with identity pivot0 column0 normalized entry1 = 13 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 · calculationBefore dual change augmented P with identity pivot0 column0 normalized entry2 := div(1, 1) After dual change augmented P with identity pivot0 column0 normalized entry2 = 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, S004 S006 · calculationBefore dual change augmented P with identity pivot0 column0 normalized entry3 := div(0, 1) After dual change augmented P with identity pivot0 column0 normalized entry3 = 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 · impliesBefore dual change augmented P with identity:pivot row=0;column=0;nonzero pivot=1;old row=[1, 13, 1, 0] After normalized full row=[1, 13, 1, 0] 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, S006 S008 · calculationBefore dual change augmented P with identity pivot1 column1 normalized entry0 := div(0, 1) After dual change augmented P with identity 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, S007 S009 · calculationBefore dual change augmented P with identity pivot1 column1 normalized entry1 := div(1, 1) After dual change augmented P with identity 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, S008 S010 · calculationBefore dual change augmented P with identity pivot1 column1 normalized entry2 := div(0, 1) After dual change augmented P with identity pivot1 column1 normalized entry2 = 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, S009 S011 · calculationBefore dual change augmented P with identity pivot1 column1 normalized entry3 := div(1, 1) After dual change augmented P with identity pivot1 column1 normalized entry3 = 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, S010 S012 · impliesBefore dual change augmented P with identity:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1, 0, 1] After normalized full row=[0, 1, 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, S011 S013 · calculationBefore dual change augmented P with identity eliminate row0 pivot1 entry0 product := mul(13, 0) After dual change augmented P with identity 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, S012 S014 · calculationBefore dual change augmented P with identity eliminate row0 pivot1 entry0 difference := sub(1, 0) After dual change augmented P with identity 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, S013 S015 · calculationBefore dual change augmented P with identity eliminate row0 pivot1 entry1 product := mul(13, 1) After dual change augmented P with identity eliminate row0 pivot1 entry1 product = 13 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, S014 S016 · calculationBefore dual change augmented P with identity eliminate row0 pivot1 entry1 difference := sub(13, 13) After dual change augmented P with identity 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, S015 S017 · calculationBefore dual change augmented P with identity eliminate row0 pivot1 entry2 product := mul(13, 0) After dual change augmented P with identity eliminate row0 pivot1 entry2 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 change augmented P with identity eliminate row0 pivot1 entry2 difference := sub(1, 0) After dual change augmented P with identity eliminate row0 pivot1 entry2 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, S017 S019 · calculationBefore dual change augmented P with identity eliminate row0 pivot1 entry3 product := mul(13, 1) After dual change augmented P with identity eliminate row0 pivot1 entry3 product = 13 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 change augmented P with identity eliminate row0 pivot1 entry3 difference := sub(0, 13) After dual change augmented P with identity eliminate row0 pivot1 entry3 difference = -13 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 · impliesBefore dual change augmented P with identity:target row=0;old=[1, 13, 1, 0];factor=13;pivot row=[0, 1, 0, 1] After new full row=[1, 0, 1, -13] 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, S020 S022 · impliesBefore dual change augmented P with identity:complete exact reduced matrix=[[1, 0, 1, -13], [0, 1, 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, S021 S023 · calculationBefore dual P times inverse row0 column0 coordinate0 product := mul(1, 1) After dual P times inverse row0 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, S022 S024 · calculationBefore dual P times inverse row0 column0 coordinate1 product := mul(13, 0) After dual P times inverse row0 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, S023 S025 · calculationBefore dual P times inverse row0 column0 complete sum := sum(1, 0) After dual P times inverse row0 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, S024 S026 · calculationBefore dual P times inverse row0 column1 coordinate0 product := mul(1, -13) After dual P times inverse row0 column1 coordinate0 product = -13 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 P times inverse row0 column1 coordinate1 product := mul(13, 1) After dual P times inverse row0 column1 coordinate1 product = 13 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 · calculationBefore dual P times inverse row0 column1 complete sum := sum(-13, 13) After dual P times inverse row0 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, S027 S029 · calculationBefore dual P times inverse row1 column0 coordinate0 product := mul(0, 1) After dual P times inverse row1 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, S028 S030 · calculationBefore dual P times inverse row1 column0 coordinate1 product := mul(1, 0) After dual P times inverse row1 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, S029 S031 · calculationBefore dual P times inverse row1 column0 complete sum := sum(0, 0) After dual P times inverse row1 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, S030 S032 · calculationBefore dual P times inverse row1 column1 coordinate0 product := mul(0, -13) After dual P times inverse row1 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, S031 S033 · calculationBefore dual P times inverse row1 column1 coordinate1 product := mul(1, 1) After dual P times inverse row1 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, S032 S034 · calculationBefore dual P times inverse row1 column1 complete sum := sum(0, 1) After dual P times inverse row1 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, S033 S035 · impliesBefore original P=[[1, 13], [0, 1]];exact row reduction [P|I] to [I|inverse];inverse=[[1, -13], [0, 1]] After complete P*inverse=identity;given old_primal=P*new_primal therefore new_primal=inverse*old_primal Conditions: none Apply the same invertible row operations to the complete augmented matrix.Accept invertibility only when the left block is I,and check every entry of P times the inverse.Solve new from the source direction old=P*new,not by multiplying the primal by P. Step identifiers and dependenciesRule: dual_invertible_change_not_same_primal_covector_law · Uses: G001, S034 S036 · calculationBefore dual new primal coordinate0 coordinate0 product := mul(1, 14) After dual new primal coordinate0 coordinate0 product = 14 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, S035 S037 · calculationBefore dual new primal coordinate0 coordinate1 product := mul(-13, 3) After dual new primal coordinate0 coordinate1 product = -39 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, S036 S038 · calculationBefore dual new primal coordinate0 complete sum := sum(14, -39) After dual new primal coordinate0 complete sum = -25 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, S037 S039 · calculationBefore dual new primal coordinate1 coordinate0 product := mul(0, 14) After dual new primal coordinate1 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, S038 S040 · calculationBefore dual new primal coordinate1 coordinate1 product := mul(1, 3) After dual new primal coordinate1 coordinate1 product = 3 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, S039 S041 · calculationBefore dual new primal coordinate1 complete sum := sum(0, 3) After dual new primal coordinate1 complete sum = 3 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, S040 S042 · calculationBefore dual new covector coordinate0 coordinate0 product := mul(2, 1) After dual new covector coordinate0 coordinate0 product = 2 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, S041 S043 · calculationBefore dual new covector coordinate0 coordinate1 product := mul(13, 0) After dual new covector coordinate0 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, S042 S044 · calculationBefore dual new covector coordinate0 complete sum := sum(2, 0) After dual new covector coordinate0 complete sum = 2 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, S043 S045 · calculationBefore dual new covector coordinate1 coordinate0 product := mul(2, 13) After dual new covector coordinate1 coordinate0 product = 26 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, S044 S046 · calculationBefore dual new covector coordinate1 coordinate1 product := mul(13, 1) After dual new covector coordinate1 coordinate1 product = 13 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, S045 S047 · calculationBefore dual new covector coordinate1 complete sum := sum(26, 13) After dual new covector coordinate1 complete sum = 39 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, S046 S048 · calculationBefore dual recover old primal coordinate0 coordinate0 product := mul(1, -25) After dual recover old primal coordinate0 coordinate0 product = -25 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, S047 S049 · calculationBefore dual recover old primal coordinate0 coordinate1 product := mul(13, 3) After dual recover old primal coordinate0 coordinate1 product = 39 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, S048 S050 · calculationBefore dual recover old primal coordinate0 complete sum := sum(-25, 39) After dual recover old primal coordinate0 complete sum = 14 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, S049 S051 · calculationBefore dual recover old primal coordinate1 coordinate0 product := mul(0, -25) After dual recover old primal coordinate1 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, S050 S052 · calculationBefore dual recover old primal coordinate1 coordinate1 product := mul(1, 3) After dual recover old primal coordinate1 coordinate1 product = 3 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, S051 S053 · calculationBefore dual recover old primal coordinate1 complete sum := sum(0, 3) After dual recover old primal coordinate1 complete sum = 3 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, S052 S054 · calculationBefore dual old covector primal pairing coordinate0 product := mul(2, 14) After dual old covector primal pairing coordinate0 product = 28 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, S053 S055 · calculationBefore dual old covector primal pairing coordinate1 product := mul(13, 3) After dual old covector primal pairing coordinate1 product = 39 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, S054 S056 · calculationBefore dual old covector primal pairing complete sum := sum(28, 39) After dual old covector primal pairing complete sum = 67 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, S055 S057 · calculationBefore dual new covector primal pairing coordinate0 product := mul(2, -25) After dual new covector primal pairing coordinate0 product = -50 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, S056 S058 · calculationBefore dual new covector primal pairing coordinate1 product := mul(39, 3) After dual new covector primal pairing coordinate1 product = 117 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, S057 S059 · calculationBefore dual new covector primal pairing complete sum := sum(-50, 117) After dual new covector primal pairing complete sum = 67 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, S058 S060 · impliesBefore original old primal=[14, 3];old covector ROW=[2, 13];given old_primal=P*new_primal After new primal=[-25, 3];new covector=old_covector*P=[2, 39];pairing preserved=67=67;canonical identification CLAIM=No Conditions: none For every new-coordinate vector,c_old*P*new=(c_old P)*new,so the covector row multiplies P on the right,opposite the inverse law for the primal.The No withholds a canonical identification without extra structure,not identifications chosen via a basis or inner product,and makes no infinite-dimensional continuous-dual claim. Step identifiers and dependenciesRule: dual_contragredient_pairing_preserved_no_canonical_identification · Uses: G001, S059 ## Reverse checks - S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S059 · premise_and_result_audit — Recompute dual new covector primal pairing complete sum with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute dual new covector primal pairing coordinate1 product with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute dual new covector primal pairing coordinate0 product with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute dual old covector primal pairing complete sum with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute dual old covector primal pairing coordinate1 product with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute dual old covector primal pairing coordinate0 product with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute dual recover old primal coordinate1 complete sum with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute dual recover old primal coordinate1 coordinate1 product with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute dual recover old primal coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute dual recover old primal coordinate0 complete sum with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute dual recover old primal coordinate0 coordinate1 product with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute dual recover old primal coordinate0 coordinate0 product with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute dual new covector coordinate1 complete sum with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute dual new covector coordinate1 coordinate1 product with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute dual new covector coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute dual new covector coordinate0 complete sum with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute dual new covector coordinate0 coordinate1 product with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute dual new covector coordinate0 coordinate0 product with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute dual new primal coordinate1 complete sum with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute dual new primal coordinate1 coordinate1 product with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute dual new primal coordinate1 coordinate0 product with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute dual new primal coordinate0 complete sum with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute dual new primal coordinate0 coordinate1 product with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute dual new primal coordinate0 coordinate0 product with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Recompute dual P times inverse row1 column1 complete sum with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute dual P times inverse row1 column1 coordinate1 product with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute dual P times inverse row1 column1 coordinate0 product with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute dual P times inverse row1 column0 complete sum with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute dual P times inverse row1 column0 coordinate1 product with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute dual P times inverse row1 column0 coordinate0 product with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute dual P times inverse row0 column1 complete sum with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute dual P times inverse row0 column1 coordinate1 product with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute dual P times inverse row0 column1 coordinate0 product with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute dual P times inverse row0 column0 complete sum with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute dual P times inverse row0 column0 coordinate1 product with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute dual P times inverse row0 column0 coordinate0 product with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Recompute dual change augmented P with identity eliminate row0 pivot1 entry3 difference with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute dual change augmented P with identity eliminate row0 pivot1 entry3 product with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute dual change augmented P with identity eliminate row0 pivot1 entry2 difference with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute dual change augmented P with identity eliminate row0 pivot1 entry2 product with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute dual change augmented P with identity eliminate row0 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute dual change augmented P with identity eliminate row0 pivot1 entry1 product with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute dual change augmented P with identity eliminate row0 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute dual change augmented P with identity eliminate row0 pivot1 entry0 product with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute dual change augmented P with identity pivot1 column1 normalized entry3 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute dual change augmented P with identity pivot1 column1 normalized entry2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute dual change augmented P with identity pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute dual change augmented P with identity pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute dual change augmented P with identity pivot0 column0 normalized entry3 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute dual change augmented P with identity pivot0 column0 normalized entry2 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute dual change augmented P with identity pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute dual change augmented P with identity 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 ([-25, 3], [2, 39], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure) 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: 78e4933c936188a6c3f547ed92164b5868d451f5e2550f3b617402bedfbeed24 - candidate record SHA-256: 0d6dcfa0ffbb70be2a2576c6f06b16c56c869ca1e1482d66a3f9d6a18760b774 - 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-0088.json)