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Question 19921: Equal Cosets Need Not Have Equal Representatives

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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R0038-19921 · q-20261014-advance-20261106-r07-quotient-vector-space-0021

Original question and givens

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 21, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[22, 2, 3], u=[24, 44, 3]. Give (difference, equivalent, criterion, boundary).

  • G001 — Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 21, 0]] of W. Coordinates are exact integers or [numerator, denominator] pairs with nonzero denominator. Quotient elements are cosets v+W, not unique representative vectors; v+W=u+W iff v−u lies in W. Membership and ranks use the whole subspace, not bounded searches over coefficients. v=[22, 2, 3], u=[24, 44, 3]. Give (difference, equivalent, criterion, 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.

Step-by-step derivation

S001 · implies

Before

source operation=quotient-coset-equivalence;original snapshot and mathematics retained

After

explicit ordered source output fields=[difference, equivalent, criterion, boundary]

Conditions: none

Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

quotient ORIGINAL ordered W basis:COMPLETE exact original ordered matrix=[[1], [21], [0]];rows=3;columns=1

After

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

Conditions: none

Entrywise invertible row operations preserve the whole real solution set,without bounding coefficients to a finite integer box.Keep original basis-column order and right-side roles rather than semantic sorting.

Step identifiers and dependencies

Rule: quotient_full_matrix_real_linear_equations_not_grid · Uses: G001, S001

S003 · calculation

Before

quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 := div(1, 1)

After

quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

quotient ORIGINAL ordered W basis:row=0;column=0;nonzero exact pivot=1;old FULL row=[1]

After

normalized FULL row=[1]

Conditions: none

Divide every entry by the same nonzero pivot,not only the left block or one entry.

Step identifiers and dependencies

Rule: quotient_nonzero_pivot_all_entry_scaling · Uses: G001, S003

S005 · calculation

Before

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product := mul(21, 1)

After

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product = 21

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference := sub(21, 21)

After

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

quotient ORIGINAL ordered W basis:target row=1;old row=[21];subtract factor=21 times FULL pivot row=[1]

After

new exact FULL row=[0]

Conditions: none

Subtract the same multiple in every entry.The inverse adds it back,preserving every equation rather than replacing the full system with sampled solutions.

Step identifiers and dependencies

Rule: quotient_invertible_complete_row_elimination · Uses: G001, S006

S008 · implies

Before

quotient ORIGINAL ordered W basis:complete exact reduced matrix=[[1], [0], [0]];ORIGINAL pivot-column indices=[0]

After

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

Conditions: none

Pivot identity coordinates prove independence and express other columns by their pivot-row coordinates,so pivot count gives rank.Invertible operations carry these relations back to the original matrix.

Step identifiers and dependencies

Rule: quotient_full_rank_and_solution_set_certificate · Uses: G001, S007

S009 · implies

Before

ACTUAL ambient real space R^3;complete ORIGINAL independent basis columns of W=[[1, 21, 0]]

After

W is the FULL real span of these columns;dim W=1;zero coset is the WHOLE subspace W,not the vector0 alone

Conditions: none

Exact rank verifies independence of the declared complete basis,whose count is dim W.The list is not the set of all W elements.An empty basis gives W={0};the zero coset is still the set W,not a representative vector.

Step identifiers and dependencies

Rule: quotient_declared_subspace_full_real_span_zero_coset · Uses: G001, S008

S010 · calculation

Before

quotient ORIGINAL v minus u coordinate0 := sub(22, 24)

After

quotient ORIGINAL v minus u coordinate0 = -2

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

quotient ORIGINAL v minus u coordinate1 := sub(2, 44)

After

quotient ORIGINAL v minus u coordinate1 = -42

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

quotient ORIGINAL v minus u coordinate2 := sub(3, 3)

After

quotient ORIGINAL v minus u coordinate2 = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

quotient coset original difference membership:COMPLETE exact original ordered matrix=[[1, -2], [21, -42], [0, 0]];rows=3;columns=2

After

perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed

Conditions: none

Entrywise invertible row operations preserve the whole real solution set,without bounding coefficients to a finite integer box.Keep original basis-column order and right-side roles rather than semantic sorting.

Step identifiers and dependencies

Rule: quotient_full_matrix_real_linear_equations_not_grid · Uses: G001, S012

S014 · calculation

Before

quotient coset original difference membership pivot0 column0 normalized entry0 := div(1, 1)

After

quotient coset original difference membership pivot0 column0 normalized entry0 = 1

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

quotient coset original difference membership pivot0 column0 normalized entry1 := div(-2, 1)

After

quotient coset original difference membership pivot0 column0 normalized entry1 = -2

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

quotient coset original difference membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, -2]

After

normalized FULL row=[1, -2]

Conditions: none

Divide every entry by the same nonzero pivot,not only the left block or one entry.

Step identifiers and dependencies

Rule: quotient_nonzero_pivot_all_entry_scaling · Uses: G001, S015

S017 · calculation

Before

quotient coset original difference membership eliminate row1 pivot0 entry0 product := mul(21, 1)

After

quotient coset original difference membership eliminate row1 pivot0 entry0 product = 21

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

quotient coset original difference membership eliminate row1 pivot0 entry0 difference := sub(21, 21)

After

quotient coset original difference membership eliminate row1 pivot0 entry0 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

quotient coset original difference membership eliminate row1 pivot0 entry1 product := mul(21, -2)

After

quotient coset original difference membership eliminate row1 pivot0 entry1 product = -42

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

quotient coset original difference membership eliminate row1 pivot0 entry1 difference := sub(-42, -42)

After

quotient coset original difference membership eliminate row1 pivot0 entry1 difference = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

quotient coset original difference membership:target row=1;old row=[21, -42];subtract factor=21 times FULL pivot row=[1, -2]

After

new exact FULL row=[0, 0]

Conditions: none

Subtract the same multiple in every entry.The inverse adds it back,preserving every equation rather than replacing the full system with sampled solutions.

Step identifiers and dependencies

Rule: quotient_invertible_complete_row_elimination · Uses: G001, S020

S022 · implies

Before

quotient coset original difference membership:column=1;remaining exact entries=[0, 0]

After

no pivot added;current rank=1

Conditions: none

Do not add a pivot for an all-zero remaining column;retain its original role and index.

Step identifiers and dependencies

Rule: quotient_zero_remaining_column_no_fake_pivot · Uses: G001, S021

S023 · implies

Before

quotient coset original difference membership:complete exact reduced matrix=[[1, -2], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0]

After

rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved

Conditions: none

Pivot identity coordinates prove independence and express other columns by their pivot-row coordinates,so pivot count gives rank.Invertible operations carry these relations back to the original matrix.

Step identifiers and dependencies

Rule: quotient_full_rank_and_solution_set_certificate · Uses: G001, S022

S024 · calculation

Before

quotient coset original difference membership original coordinate0 reconstruction coefficient0 product := mul(1, -2)

After

quotient coset original difference membership original coordinate0 reconstruction coefficient0 product = -2

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

quotient coset original difference membership original coordinate0 reconstruction complete sum := sum(-2)

After

quotient coset original difference membership original coordinate0 reconstruction complete sum = -2

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

quotient coset original difference membership original coordinate1 reconstruction coefficient0 product := mul(21, -2)

After

quotient coset original difference membership original coordinate1 reconstruction coefficient0 product = -42

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

quotient coset original difference membership original coordinate1 reconstruction complete sum := sum(-42)

After

quotient coset original difference membership original coordinate1 reconstruction complete sum = -42

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

quotient coset original difference membership original coordinate2 reconstruction coefficient0 product := mul(0, -2)

After

quotient coset original difference membership original coordinate2 reconstruction coefficient0 product = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

quotient coset original difference membership original coordinate2 reconstruction complete sum := sum(0)

After

quotient coset original difference membership original coordinate2 reconstruction complete sum = 0

Conditions: none

Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

quotient coset original difference membership:original proposed vector=[-2, -42, 0];original ordered independent columns=[[1, 21, 0]];complete reduced augmented system=[[1, -2], [0, 0], [0, 0]]

After

exact real-span coefficient witness=[-2];FULL original reconstruction=[-2, -42, 0];membership=Yes

Conditions: none

Read exact coefficients from the pivot equations and reconstruct every original coordinate.Empty coefficients reconstruct only the zero vector for an empty basis.Valid coefficients for nonempty bases may be fractional or outside original input bounds and must not be excluded.

Step identifiers and dependencies

Rule: quotient_full_span_coefficient_witness_not_sampled_membership · Uses: G001, S029

S031 · implies

Before

ORIGINAL v=[22, 2, 3],u=[24, 44, 3];ORDERED difference v-u=[-2, -42, 0];full-real-span membership=Yes

After

v+W=u+W iff v-u in W;coset equality=Yes;not a test of representative vector equality

Conditions: none

If the cosets agree,v=v+0 belongs to u+W,so v−u lies in W.Conversely,if v−u lies in W,then v+w=u+(v−u+w)belongs to u+W,and the negative difference proves the reverse inclusion.The difference need not be0;representative vectors need not agree.

Step identifiers and dependencies

Rule: quotient_coset_equality_both_inclusions_not_representative_equality · Uses: G001, S030

Reverse checks

  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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 quotient coset original difference membership eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute quotient coset original difference membership eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute quotient coset original difference membership eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute quotient coset original difference membership eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Recompute quotient coset original difference membership pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute quotient coset original difference membership pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals.
  • 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 quotient ORIGINAL v minus u coordinate2 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute quotient ORIGINAL v minus u coordinate1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute quotient ORIGINAL v minus u coordinate0 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Recompute quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Recompute quotient ORIGINAL ordered W 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

([-2, -42, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

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: 8c732b9fe01015313af90605481d6a094a1c7268ea56fc172d5f71ffcac09758
  • candidate record SHA-256: fe96e0f8d7524890aa07e1c7df9dd98118051e58ca22a5074eb7529e07278fd2
  • semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612
  • question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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