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Question 18727: Every Dual-Basis Pairing Must Be a Kronecker Delta

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0036-18727 · q-20261014-advance-20261105-r02-dual-vector-space-0027

Original question and givens

Use ordered basis column vectors=[[1, 0], [2, 1]] and candidate dual rows=[[1, -2], [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], [2, 1]] and candidate dual rows=[[1, -2], [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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

dual original ordered basis:complete ordered matrix=[[1, 2], [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 dependencies

Rule: finite_linear_source_matrix_not_semantic_reordering · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

dual original ordered basis pivot0 column0 normalized entry1 := div(2, 1)

After

dual original ordered basis pivot0 column0 normalized entry1 = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

dual original ordered basis:pivot row=0;column=0;nonzero pivot=1;old row=[1, 2]

After

normalized full row=[1, 2]

Conditions: none

Divide every entry by the same nonzero pivot,retaining exact rational zero values and signs.

Step identifiers and dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S004

S006 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: finite_linear_nonzero_pivot_complete_row_scaling · Uses: G001, S007

S009 · calculation

Before

dual original ordered basis eliminate row0 pivot1 entry0 product := mul(2, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

dual original ordered basis eliminate row0 pivot1 entry1 product := mul(2, 1)

After

dual original ordered basis eliminate row0 pivot1 entry1 product = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

dual original ordered basis eliminate row0 pivot1 entry1 difference := sub(2, 2)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

dual original ordered basis:target row=0;old=[1, 2];factor=2;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 dependencies

Rule: finite_linear_invertible_all_entry_elimination · Uses: G001, S012

S014 · implies

Before

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 dependencies

Rule: finite_linear_complete_rank_certificate_not_samples · Uses: G001, S013

S015 · implies

Before

dual original ordered basis:complete ORIGINAL independent columns=[[1, 0], [2, 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 dependencies

Rule: directsum_declared_basis_complete_real_span · Uses: G001, S014

S016 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

dual candidate row0 basis column0 coordinate1 product := mul(-2, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

dual candidate row0 basis column1 coordinate0 product := mul(1, 2)

After

dual candidate row0 basis column1 coordinate0 product = 2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

dual candidate row0 basis column1 coordinate1 product := mul(-2, 1)

After

dual candidate row0 basis column1 coordinate1 product = -2

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

dual candidate row0 basis column1 complete sum := sum(2, -2)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

dual candidate row1 basis column1 coordinate0 product := mul(0, 2)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

ORIGINAL basis columns=[[1, 0], [2, 1]];candidate functional ROWS=[[1, -2], [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 dependencies

Rule: 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: b81ae983da895c21e992510fa8eed7791815120907714ebb01fbbad6f750a0a8
  • candidate record SHA-256: 9b39489a601ec82d2ff4193ac44b33042a0c40d95e26ce4f000d60529bc2a182
  • semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301
  • question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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