# Question 19937: Report Representatives of Quotient Operations, Not Unique Coset Names

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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.

R0038-19937 · q-20261014-advance-20261106-r07-quotient-vector-space-0037



## Original question and givens

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 12, 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=[13, 2, 3], u=[3, 12, 4], scalar=12. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).


- G001 — Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 12, 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=[13, 2, 3], u=[3, 12, 4], scalar=12. Give (sum_representative, scalar_representative, addition_law, scalar_law, 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-quotient-vector-space](https://ocme.evemisslab.com/en/mko/mko-quotient-vector-space/)





## Step-by-step derivation

S001 · impliesBefore

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

After

explicit ordered source output fields=[sum_representative, scalar_representative, addition_law, scalar_law, 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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

quotient ORIGINAL ordered W basis:COMPLETE exact original ordered matrix=[[1], [12], [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 dependenciesRule: quotient_full_matrix_real_linear_equations_not_grid · Uses: G001, S001





S003 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

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 dependenciesRule: quotient_nonzero_pivot_all_entry_scaling · Uses: G001, S003





S005 · calculationBefore

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product := mul(12, 1)

After

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product = 12

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference := sub(12, 12)

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

quotient ORIGINAL ordered W basis:target row=1;old row=[12];subtract factor=12 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 dependenciesRule: quotient_invertible_complete_row_elimination · Uses: G001, S006





S008 · impliesBefore

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 dependenciesRule: quotient_full_rank_and_solution_set_certificate · Uses: G001, S007





S009 · impliesBefore

ACTUAL ambient real space R^3;complete ORIGINAL independent basis columns of W=[[1, 12, 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 dependenciesRule: quotient_declared_subspace_full_real_span_zero_coset · Uses: G001, S008





S010 · impliesBefore

ACTUAL subspace W full real span in R^3;original basis=[[1, 12, 0]]

After

for ANY representatives v'=v+w,u'=u+z with w,z in W,and ANY real lambda,sum difference=w+z in W and scalar difference=lambda*w in W;thus quotient addition/scaling independent of representatives

Conditions: none

Subspace closure under addition,negation and every real scalar keeps all representative-change differences in W.This proves well-definedness for arbitrary changes,not just supplied shifts.Quotient objects are cosets:representatives are generally nonunique,with uniqueness possible when W={0},without equating the coset object type with vectors.

Step identifiers and dependenciesRule: quotient_arbitrary_representatives_well_defined_both_operations · Uses: G001, S009





S011 · calculationBefore

quotient sum representative coordinate0 := sum(13, 3)

After

quotient sum representative coordinate0 = 16

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

quotient sum representative coordinate1 := sum(2, 12)

After

quotient sum representative coordinate1 = 14

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

quotient sum representative coordinate2 := sum(3, 4)

After

quotient sum representative coordinate2 = 7

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

quotient scaled representative coordinate0 := mul(12, 13)

After

quotient scaled representative coordinate0 = 156

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · calculationBefore

quotient scaled representative coordinate1 := mul(12, 2)

After

quotient scaled representative coordinate1 = 24

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

quotient scaled representative coordinate2 := mul(12, 3)

After

quotient scaled representative coordinate2 = 36

Conditions: none

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

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · impliesBefore

ORIGINAL representatives v=[13, 2, 3],u=[3, 12, 4];original real scalar=12

After

reported representative vectors v+u=[16, 14, 7],lambda*v=[156, 24, 36];resulting objects are their cosets,not a selected generic unique vector name

Conditions: none

Keep the exact representative vectors in the first fields and interpret them through the quotient laws.No canonical representative selector is introduced;the zero quotient element remains the entire W.

Step identifiers and dependenciesRule: quotient_operation_representatives_not_coset_object_identity · Uses: G001, S016









## Reverse checks


- S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S016 · premise_and_result_audit — Recompute quotient scaled representative coordinate2 with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Recompute quotient scaled representative coordinate1 with exact rational arithmetic, not rounded decimals.
- S014 · premise_and_result_audit — Recompute quotient scaled representative coordinate0 with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Recompute quotient sum representative coordinate2 with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute quotient sum representative coordinate1 with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute quotient sum representative coordinate0 with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

([16, 14, 7], [156, 24, 36], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

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: 07e3821523f8bd696141d3ef9680cdb68a5ad69c35ed4e454c154bd75fd9e9c2
- candidate record SHA-256: 65385cccd81ee2ba93aa7d2df3b1cbe7d27268ff813fe1bf9e6dad09a66b9914
- semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612
- question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7
- Lean: NOT_RUN · human review: NOT_PERFORMED

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