Question 18703: Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Original question and givens

Use invertible change matrix P=[[1, 3], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[4, 3] and old covector row=[2, 3]. 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, 3], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[4, 3] and old covector row=[2, 3]. 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

Step-by-step derivation

source operation=dual-coordinate-change-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary]

dual change augmented P with identity:complete ordered matrix=[[1, 3, 1, 0], [0, 1, 0, 1]];rows=2;columns=4

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

dual change augmented P with identity pivot0 column0 normalized entry0 := div(1, 1)

dual change augmented P with identity pivot0 column0 normalized entry0 = 1

dual change augmented P with identity pivot0 column0 normalized entry1 := div(3, 1)

dual change augmented P with identity pivot0 column0 normalized entry1 = 3

dual change augmented P with identity pivot0 column0 normalized entry2 := div(1, 1)

dual change augmented P with identity pivot0 column0 normalized entry2 = 1

dual change augmented P with identity pivot0 column0 normalized entry3 := div(0, 1)

dual change augmented P with identity pivot0 column0 normalized entry3 = 0

dual change augmented P with identity:pivot row=0;column=0;nonzero pivot=1;old row=[1, 3, 1, 0]

normalized full row=[1, 3, 1, 0]

dual change augmented P with identity pivot1 column1 normalized entry0 := div(0, 1)

dual change augmented P with identity pivot1 column1 normalized entry0 = 0

dual change augmented P with identity pivot1 column1 normalized entry1 := div(1, 1)

dual change augmented P with identity pivot1 column1 normalized entry1 = 1

dual change augmented P with identity pivot1 column1 normalized entry2 := div(0, 1)

dual change augmented P with identity pivot1 column1 normalized entry2 = 0

dual change augmented P with identity pivot1 column1 normalized entry3 := div(1, 1)

dual change augmented P with identity pivot1 column1 normalized entry3 = 1

dual change augmented P with identity:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1, 0, 1]

normalized full row=[0, 1, 0, 1]

dual change augmented P with identity eliminate row0 pivot1 entry0 product := mul(3, 0)

dual change augmented P with identity eliminate row0 pivot1 entry0 product = 0

dual change augmented P with identity eliminate row0 pivot1 entry0 difference := sub(1, 0)

dual change augmented P with identity eliminate row0 pivot1 entry0 difference = 1

dual change augmented P with identity eliminate row0 pivot1 entry1 product := mul(3, 1)

dual change augmented P with identity eliminate row0 pivot1 entry1 product = 3

dual change augmented P with identity eliminate row0 pivot1 entry1 difference := sub(3, 3)

dual change augmented P with identity eliminate row0 pivot1 entry1 difference = 0

dual change augmented P with identity eliminate row0 pivot1 entry2 product := mul(3, 0)

dual change augmented P with identity eliminate row0 pivot1 entry2 product = 0

dual change augmented P with identity eliminate row0 pivot1 entry2 difference := sub(1, 0)

dual change augmented P with identity eliminate row0 pivot1 entry2 difference = 1

dual change augmented P with identity eliminate row0 pivot1 entry3 product := mul(3, 1)

dual change augmented P with identity eliminate row0 pivot1 entry3 product = 3

dual change augmented P with identity eliminate row0 pivot1 entry3 difference := sub(0, 3)

dual change augmented P with identity eliminate row0 pivot1 entry3 difference = -3

dual change augmented P with identity:target row=0;old=[1, 3, 1, 0];factor=3;pivot row=[0, 1, 0, 1]

new full row=[1, 0, 1, -3]

dual change augmented P with identity:complete exact reduced matrix=[[1, 0, 1, -3], [0, 1, 0, 1]];pivot columns=[0, 1]

rank=2;pivot columns independent and all original column relations preserved;zero rows retained

dual P times inverse row0 column0 coordinate0 product := mul(1, 1)

dual P times inverse row0 column0 coordinate0 product = 1

dual P times inverse row0 column0 coordinate1 product := mul(3, 0)

dual P times inverse row0 column0 coordinate1 product = 0

dual P times inverse row0 column0 complete sum := sum(1, 0)

dual P times inverse row0 column0 complete sum = 1

dual P times inverse row0 column1 coordinate0 product := mul(1, -3)

dual P times inverse row0 column1 coordinate0 product = -3

dual P times inverse row0 column1 coordinate1 product := mul(3, 1)

dual P times inverse row0 column1 coordinate1 product = 3

dual P times inverse row0 column1 complete sum := sum(-3, 3)

dual P times inverse row0 column1 complete sum = 0

dual P times inverse row1 column0 coordinate0 product := mul(0, 1)

dual P times inverse row1 column0 coordinate0 product = 0

dual P times inverse row1 column0 coordinate1 product := mul(1, 0)

dual P times inverse row1 column0 coordinate1 product = 0

dual P times inverse row1 column0 complete sum := sum(0, 0)

dual P times inverse row1 column0 complete sum = 0

dual P times inverse row1 column1 coordinate0 product := mul(0, -3)

dual P times inverse row1 column1 coordinate0 product = 0

dual P times inverse row1 column1 coordinate1 product := mul(1, 1)

dual P times inverse row1 column1 coordinate1 product = 1

dual P times inverse row1 column1 complete sum := sum(0, 1)

dual P times inverse row1 column1 complete sum = 1

original P=[[1, 3], [0, 1]];exact row reduction [P|I] to [I|inverse];inverse=[[1, -3], [0, 1]]

complete P*inverse=identity;given old_primal=P*new_primal therefore new_primal=inverse*old_primal

dual new primal coordinate0 coordinate0 product := mul(1, 4)

dual new primal coordinate0 coordinate0 product = 4

dual new primal coordinate0 coordinate1 product := mul(-3, 3)

dual new primal coordinate0 coordinate1 product = -9

dual new primal coordinate0 complete sum := sum(4, -9)

dual new primal coordinate0 complete sum = -5

dual new primal coordinate1 coordinate0 product := mul(0, 4)

dual new primal coordinate1 coordinate0 product = 0

dual new primal coordinate1 coordinate1 product := mul(1, 3)

dual new primal coordinate1 coordinate1 product = 3

dual new primal coordinate1 complete sum := sum(0, 3)

dual new primal coordinate1 complete sum = 3

dual new covector coordinate0 coordinate0 product := mul(2, 1)

dual new covector coordinate0 coordinate0 product = 2

dual new covector coordinate0 coordinate1 product := mul(3, 0)

dual new covector coordinate0 coordinate1 product = 0

dual new covector coordinate0 complete sum := sum(2, 0)

dual new covector coordinate0 complete sum = 2

dual new covector coordinate1 coordinate0 product := mul(2, 3)

dual new covector coordinate1 coordinate0 product = 6

dual new covector coordinate1 coordinate1 product := mul(3, 1)

dual new covector coordinate1 coordinate1 product = 3

dual new covector coordinate1 complete sum := sum(6, 3)

dual new covector coordinate1 complete sum = 9

dual recover old primal coordinate0 coordinate0 product := mul(1, -5)

dual recover old primal coordinate0 coordinate0 product = -5

dual recover old primal coordinate0 coordinate1 product := mul(3, 3)

dual recover old primal coordinate0 coordinate1 product = 9

dual recover old primal coordinate0 complete sum := sum(-5, 9)

dual recover old primal coordinate0 complete sum = 4

dual recover old primal coordinate1 coordinate0 product := mul(0, -5)

dual recover old primal coordinate1 coordinate0 product = 0

dual recover old primal coordinate1 coordinate1 product := mul(1, 3)

dual recover old primal coordinate1 coordinate1 product = 3

dual recover old primal coordinate1 complete sum := sum(0, 3)

dual recover old primal coordinate1 complete sum = 3

dual old covector primal pairing coordinate0 product := mul(2, 4)

dual old covector primal pairing coordinate0 product = 8

dual old covector primal pairing coordinate1 product := mul(3, 3)

dual old covector primal pairing coordinate1 product = 9

dual old covector primal pairing complete sum := sum(8, 9)

dual old covector primal pairing complete sum = 17

dual new covector primal pairing coordinate0 product := mul(2, -5)

dual new covector primal pairing coordinate0 product = -10

dual new covector primal pairing coordinate1 product := mul(9, 3)

dual new covector primal pairing coordinate1 product = 27

dual new covector primal pairing complete sum := sum(-10, 27)

dual new covector primal pairing complete sum = 17

original old primal=[4, 3];old covector ROW=[2, 3];given old_primal=P*new_primal

new primal=[-5, 3];new covector=old_covector*P=[2, 9];pairing preserved=17=17;canonical identification CLAIM=No

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

([-5, 3], [2, 9], 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.