# Coset Equality, Representative Independence and Finite Quotient Dimension

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261106-r07-quotient-vector-space-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 1, 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=[2, 2, 3], u=[4, 4, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -2, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -2, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0001/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 2. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 2, 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=[3, 2, 3], u=[5, 6, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -4, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -4, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0002/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 3. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 3, 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=[4, 2, 3], u=[6, 8, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -6, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -6, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0003/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 4. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 4, 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=[5, 2, 3], u=[7, 10, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -8, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -8, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0004/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 5. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 5, 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=[6, 2, 3], u=[8, 12, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -10, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -10, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0005/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 6. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 6, 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=[7, 2, 3], u=[9, 14, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -12, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -12, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0006/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 7. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 7, 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=[8, 2, 3], u=[10, 16, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -14, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -14, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0007/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 8. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 8, 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=[9, 2, 3], u=[11, 18, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -16, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -16, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0008/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 9. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 9, 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=[10, 2, 3], u=[12, 20, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -18, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -18, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0009/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 10. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 10, 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=[11, 2, 3], u=[13, 22, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -20, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -20, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0010/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 11. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 11, 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=[12, 2, 3], u=[14, 24, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -22, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -22, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0011/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 12. Equal Cosets Need Not Have Equal Representatives

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=[15, 26, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -24, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -24, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0012/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 13. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 13, 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=[14, 2, 3], u=[16, 28, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -26, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -26, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0013/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 14. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 14, 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=[15, 2, 3], u=[17, 30, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -28, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -28, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0014/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 15. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 15, 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=[16, 2, 3], u=[18, 32, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -30, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -30, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0015/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 16. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 16, 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=[17, 2, 3], u=[19, 34, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -32, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -32, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0016/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 17. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 17, 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=[18, 2, 3], u=[20, 36, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -34, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -34, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0017/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 18. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 18, 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=[19, 2, 3], u=[21, 38, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -36, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -36, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0018/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 19. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 19, 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=[20, 2, 3], u=[22, 40, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -38, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -38, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0019/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 20. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 20, 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=[21, 2, 3], u=[23, 42, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -40, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -40, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0020/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 21. Equal Cosets Need Not Have Equal Representatives

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).

View answer and explanation([-2, -42, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -42, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0021/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 22. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 22, 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=[23, 2, 3], u=[25, 46, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -44, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -44, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0022/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 23. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 23, 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=[24, 2, 3], u=[26, 48, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -46, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -46, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0023/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 24. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 24, 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=[25, 2, 3], u=[27, 50, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -48, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -48, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0024/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 25. Equal Cosets Need Not Have Equal Representatives

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 25, 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=[26, 2, 3], u=[28, 52, 3]. Give (difference, equivalent, criterion, boundary).

View answer and explanation([-2, -50, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives)

Compute the ordered difference v−u and check its membership in W. Equal representatives imply equal cosets, but a nonzero difference in W also gives equality; no unique representative is selected. The exact result, retaining the requested order, is ([-2, -50, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0025/)

Quotient vector space · qf-mod37-quotient-coset-equivalence · candidate





## 26. Report Representatives of Quotient Operations, Not Unique Coset Names

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

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([5, 3, 7], [2, 2, 3], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([5, 3, 7], [2, 2, 3], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0026/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 27. Report Representatives of Quotient Operations, Not Unique Coset Names

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

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([6, 4, 7], [6, 4, 6], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([6, 4, 7], [6, 4, 6], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0027/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 28. Report Representatives of Quotient Operations, Not Unique Coset Names

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

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([7, 5, 7], [12, 6, 9], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([7, 5, 7], [12, 6, 9], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0028/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 29. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 4, 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=[5, 2, 3], u=[3, 4, 4], scalar=4. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([8, 6, 7], [20, 8, 12], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([8, 6, 7], [20, 8, 12], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0029/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 30. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 5, 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=[6, 2, 3], u=[3, 5, 4], scalar=5. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([9, 7, 7], [30, 10, 15], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([9, 7, 7], [30, 10, 15], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0030/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 31. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 6, 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=[7, 2, 3], u=[3, 6, 4], scalar=6. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([10, 8, 7], [42, 12, 18], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([10, 8, 7], [42, 12, 18], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0031/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 32. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 7, 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=[8, 2, 3], u=[3, 7, 4], scalar=7. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([11, 9, 7], [56, 14, 21], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([11, 9, 7], [56, 14, 21], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0032/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 33. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 8, 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=[9, 2, 3], u=[3, 8, 4], scalar=8. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([12, 10, 7], [72, 16, 24], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([12, 10, 7], [72, 16, 24], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0033/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 34. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 9, 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=[10, 2, 3], u=[3, 9, 4], scalar=9. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([13, 11, 7], [90, 18, 27], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([13, 11, 7], [90, 18, 27], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0034/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 35. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 10, 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=[11, 2, 3], u=[3, 10, 4], scalar=10. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([14, 12, 7], [110, 20, 30], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([14, 12, 7], [110, 20, 30], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0035/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 36. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 11, 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=[12, 2, 3], u=[3, 11, 4], scalar=11. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([15, 13, 7], [132, 22, 33], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([15, 13, 7], [132, 22, 33], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0036/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 37. Report Representatives of Quotient Operations, Not Unique Coset Names

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).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

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

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([16, 14, 7], [156, 24, 36], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0037/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 38. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 13, 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=[14, 2, 3], u=[3, 13, 4], scalar=13. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([17, 15, 7], [182, 26, 39], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([17, 15, 7], [182, 26, 39], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0038/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 39. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 14, 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=[15, 2, 3], u=[3, 14, 4], scalar=14. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([18, 16, 7], [210, 28, 42], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([18, 16, 7], [210, 28, 42], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0039/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 40. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 15, 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=[16, 2, 3], u=[3, 15, 4], scalar=15. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([19, 17, 7], [240, 30, 45], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([19, 17, 7], [240, 30, 45], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0040/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 41. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 16, 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=[17, 2, 3], u=[3, 16, 4], scalar=16. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([20, 18, 7], [272, 32, 48], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([20, 18, 7], [272, 32, 48], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0041/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 42. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 17, 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=[18, 2, 3], u=[3, 17, 4], scalar=17. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([21, 19, 7], [306, 34, 51], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([21, 19, 7], [306, 34, 51], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0042/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 43. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 18, 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=[19, 2, 3], u=[3, 18, 4], scalar=18. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([22, 20, 7], [342, 36, 54], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([22, 20, 7], [342, 36, 54], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0043/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 44. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 19, 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=[20, 2, 3], u=[3, 19, 4], scalar=19. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([23, 21, 7], [380, 38, 57], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([23, 21, 7], [380, 38, 57], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0044/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 45. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 20, 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=[21, 2, 3], u=[3, 20, 4], scalar=20. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([24, 22, 7], [420, 40, 60], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([24, 22, 7], [420, 40, 60], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0045/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 46. Report Representatives of Quotient Operations, Not Unique Coset Names

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=[3, 21, 4], scalar=21. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([25, 23, 7], [462, 42, 63], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([25, 23, 7], [462, 42, 63], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0046/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 47. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 22, 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=[23, 2, 3], u=[3, 22, 4], scalar=22. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([26, 24, 7], [506, 44, 66], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([26, 24, 7], [506, 44, 66], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0047/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 48. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 23, 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=[24, 2, 3], u=[3, 23, 4], scalar=23. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([27, 25, 7], [552, 46, 69], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([27, 25, 7], [552, 46, 69], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0048/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 49. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 24, 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=[25, 2, 3], u=[3, 24, 4], scalar=24. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([28, 26, 7], [600, 48, 72], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([28, 26, 7], [600, 48, 72], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0049/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 50. Report Representatives of Quotient Operations, Not Unique Coset Names

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 25, 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=[26, 2, 3], u=[3, 25, 4], scalar=25. Give (sum_representative, scalar_representative, addition_law, scalar_law, boundary).

The first two fields are representative vectors, not unique vector names for quotient elements; the zero coset is the whole subspace W.

View answer and explanation([29, 27, 7], [650, 50, 75], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements)

Return representative vectors v+u and scalar*v with the two quotient laws. They name resulting cosets but are not unique vector-valued quotient elements; the zero coset is W itself. The exact result, retaining the requested order, is ([29, 27, 7], [650, 50, 75], [v]+[u]=[v+u], lambda[v]=[lambda v], representatives_only_not_unique_quotient_elements).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0050/)

Quotient vector space · qf-mod37-quotient-coset-operations · candidate





## 51. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 1, 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=[2, 2, 3], u=[3, 1, 4], shift_v_coefficients=[2], shift_u_coefficients=[-1] in supplied basis order, scalar=1. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0051/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 52. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 2, 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=[3, 2, 3], u=[3, 2, 4], shift_v_coefficients=[3], shift_u_coefficients=[-2] in supplied basis order, scalar=2. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0052/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 53. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 3, 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=[4, 2, 3], u=[3, 3, 4], shift_v_coefficients=[4], shift_u_coefficients=[-3] in supplied basis order, scalar=3. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0053/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 54. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 4, 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=[5, 2, 3], u=[3, 4, 4], shift_v_coefficients=[5], shift_u_coefficients=[-4] in supplied basis order, scalar=4. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0054/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 55. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 5, 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=[6, 2, 3], u=[3, 5, 4], shift_v_coefficients=[6], shift_u_coefficients=[-5] in supplied basis order, scalar=5. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0055/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 56. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 6, 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=[7, 2, 3], u=[3, 6, 4], shift_v_coefficients=[7], shift_u_coefficients=[-6] in supplied basis order, scalar=6. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0056/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 57. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 7, 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=[8, 2, 3], u=[3, 7, 4], shift_v_coefficients=[8], shift_u_coefficients=[-7] in supplied basis order, scalar=7. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0057/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 58. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 8, 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=[9, 2, 3], u=[3, 8, 4], shift_v_coefficients=[9], shift_u_coefficients=[-8] in supplied basis order, scalar=8. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0058/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 59. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 9, 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=[10, 2, 3], u=[3, 9, 4], shift_v_coefficients=[10], shift_u_coefficients=[-9] in supplied basis order, scalar=9. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0059/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 60. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 10, 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=[11, 2, 3], u=[3, 10, 4], shift_v_coefficients=[11], shift_u_coefficients=[-10] in supplied basis order, scalar=10. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0060/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 61. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 11, 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=[12, 2, 3], u=[3, 11, 4], shift_v_coefficients=[12], shift_u_coefficients=[-11] in supplied basis order, scalar=11. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0061/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 62. Changing Representatives by W Must Preserve Both Operations

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], shift_v_coefficients=[13], shift_u_coefficients=[-12] in supplied basis order, scalar=12. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0062/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 63. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 13, 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=[14, 2, 3], u=[3, 13, 4], shift_v_coefficients=[14], shift_u_coefficients=[-13] in supplied basis order, scalar=13. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0063/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 64. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 14, 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=[15, 2, 3], u=[3, 14, 4], shift_v_coefficients=[15], shift_u_coefficients=[-14] in supplied basis order, scalar=14. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0064/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 65. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 15, 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=[16, 2, 3], u=[3, 15, 4], shift_v_coefficients=[16], shift_u_coefficients=[-15] in supplied basis order, scalar=15. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0065/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 66. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 16, 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=[17, 2, 3], u=[3, 16, 4], shift_v_coefficients=[17], shift_u_coefficients=[-16] in supplied basis order, scalar=16. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0066/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 67. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 17, 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=[18, 2, 3], u=[3, 17, 4], shift_v_coefficients=[18], shift_u_coefficients=[-17] in supplied basis order, scalar=17. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0067/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 68. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 18, 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=[19, 2, 3], u=[3, 18, 4], shift_v_coefficients=[19], shift_u_coefficients=[-18] in supplied basis order, scalar=18. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0068/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 69. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 19, 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=[20, 2, 3], u=[3, 19, 4], shift_v_coefficients=[20], shift_u_coefficients=[-19] in supplied basis order, scalar=19. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0069/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 70. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 20, 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=[21, 2, 3], u=[3, 20, 4], shift_v_coefficients=[21], shift_u_coefficients=[-20] in supplied basis order, scalar=20. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0070/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 71. Changing Representatives by W Must Preserve Both Operations

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=[3, 21, 4], shift_v_coefficients=[22], shift_u_coefficients=[-21] in supplied basis order, scalar=21. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0071/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 72. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 22, 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=[23, 2, 3], u=[3, 22, 4], shift_v_coefficients=[23], shift_u_coefficients=[-22] in supplied basis order, scalar=22. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0072/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 73. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 23, 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=[24, 2, 3], u=[3, 23, 4], shift_v_coefficients=[24], shift_u_coefficients=[-23] in supplied basis order, scalar=23. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0073/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 74. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 24, 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=[25, 2, 3], u=[3, 24, 4], shift_v_coefficients=[25], shift_u_coefficients=[-24] in supplied basis order, scalar=24. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0074/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 75. Changing Representatives by W Must Preserve Both Operations

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 25, 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=[26, 2, 3], u=[3, 25, 4], shift_v_coefficients=[26], shift_u_coefficients=[-25] in supplied basis order, scalar=25. Give (shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate).

View answer and explanation(Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives)

Form both W shifts from their full basis coefficients. Check the two shifts, the changed-sum difference and the changed-scaled-v difference all lie in W. These exact membership obligations express representative independence, not a new Lean proof. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_representatives).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0075/)

Quotient vector space · qf-mod37-quotient-representative-independence · candidate





## 76. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 1, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0076/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 77. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 2, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0077/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 78. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 3, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0078/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 79. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 4, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0079/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 80. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 5, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0080/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 81. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 6, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0081/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 82. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 7, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0082/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 83. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 8, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0083/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 84. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 9, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0084/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 85. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 10, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0085/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 86. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 11, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0086/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 87. Finite Quotient Dimension Subtracts Subspace Dimension

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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0087/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 88. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 13, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0088/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 89. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 14, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0089/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 90. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 15, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0090/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 91. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 16, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0091/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 92. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 17, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0092/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 93. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 18, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0093/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 94. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 19, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0094/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 95. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 20, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0095/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 96. Finite Quotient Dimension Subtracts Subspace Dimension

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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0096/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 97. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 22, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0097/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 98. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 23, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0098/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 99. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 24, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0099/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate





## 100. Finite Quotient Dimension Subtracts Subspace Dimension

Use V=R^3 and the complete independent ordered column-vector basis basis_w=[[1, 25, 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate).

View answer and explanation(3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W)

Count the complete independent W basis and compute ambient dimension minus that count. The formula is finite-dimensional; W={0} leaves the ambient dimension, W=V gives dimension0, and the zero quotient element is the whole coset W. The exact result, retaining the requested order, is (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0100/)

Quotient vector space · qf-mod37-quotient-dimension-formula · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261106-r07-quotient-vector-space-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261106-r07-quotient-vector-space-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261106-r07-quotient-vector-space-foundations/validation-report.json)