Coset Equality, Representative Independence and Finite Quotient Dimension
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Quotient vector space · qf-mod37-quotient-dimension-formula · candidate
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