# Question 19910: Equal Cosets Need Not Have Equal Representatives [Return to catalogue page 100](https://ocme.evemisslab.com/en/questions/page/100/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261106-r07-quotient-vector-space-foundations/#q-20261014-advance-20261106-r07-quotient-vector-space-0010) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0038-19910 · q-20261014-advance-20261106-r07-quotient-vector-space-0010 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-quotient-vector-space](https://ocme.evemisslab.com/en/mko/mko-quotient-vector-space/) ## Step-by-step derivation S001 · impliesBefore source operation=quotient-coset-equivalence;original snapshot and mathematics retained After explicit ordered source output fields=[difference, equivalent, criterion, boundary] Conditions: none Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore quotient ORIGINAL ordered W basis:COMPLETE exact original ordered matrix=[[1], [10], [0]];rows=3;columns=1 After perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed Conditions: none Entrywise invertible row operations preserve the whole real solution set,without bounding coefficients to a finite integer box.Keep original basis-column order and right-side roles rather than semantic sorting. Step identifiers and dependenciesRule: quotient_full_matrix_real_linear_equations_not_grid · Uses: G001, S001 S003 · calculationBefore quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 := div(1, 1) After quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore quotient ORIGINAL ordered W basis:row=0;column=0;nonzero exact pivot=1;old FULL row=[1] After normalized FULL row=[1] Conditions: none Divide every entry by the same nonzero pivot,not only the left block or one entry. Step identifiers and dependenciesRule: quotient_nonzero_pivot_all_entry_scaling · Uses: G001, S003 S005 · calculationBefore quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product := mul(10, 1) After quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product = 10 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference := sub(10, 10) After quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore quotient ORIGINAL ordered W basis:target row=1;old row=[10];subtract factor=10 times FULL pivot row=[1] After new exact FULL row=[0] Conditions: none Subtract the same multiple in every entry.The inverse adds it back,preserving every equation rather than replacing the full system with sampled solutions. Step identifiers and dependenciesRule: quotient_invertible_complete_row_elimination · Uses: G001, S006 S008 · impliesBefore quotient ORIGINAL ordered W basis:complete exact reduced matrix=[[1], [0], [0]];ORIGINAL pivot-column indices=[0] After rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved Conditions: none Pivot identity coordinates prove independence and express other columns by their pivot-row coordinates,so pivot count gives rank.Invertible operations carry these relations back to the original matrix. Step identifiers and dependenciesRule: quotient_full_rank_and_solution_set_certificate · Uses: G001, S007 S009 · impliesBefore ACTUAL ambient real space R^3;complete ORIGINAL independent basis columns of W=[[1, 10, 0]] After W is the FULL real span of these columns;dim W=1;zero coset is the WHOLE subspace W,not the vector0 alone Conditions: none Exact rank verifies independence of the declared complete basis,whose count is dim W.The list is not the set of all W elements.An empty basis gives W={0};the zero coset is still the set W,not a representative vector. Step identifiers and dependenciesRule: quotient_declared_subspace_full_real_span_zero_coset · Uses: G001, S008 S010 · calculationBefore quotient ORIGINAL v minus u coordinate0 := sub(11, 13) After quotient ORIGINAL v minus u coordinate0 = -2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore quotient ORIGINAL v minus u coordinate1 := sub(2, 22) After quotient ORIGINAL v minus u coordinate1 = -20 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore quotient ORIGINAL v minus u coordinate2 := sub(3, 3) After quotient ORIGINAL v minus u coordinate2 = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore quotient coset original difference membership:COMPLETE exact original ordered matrix=[[1, -2], [10, -20], [0, 0]];rows=3;columns=2 After perform invertible ROW operations on EVERY entry;original COLUMN and right-side roles remain fixed Conditions: none Entrywise invertible row operations preserve the whole real solution set,without bounding coefficients to a finite integer box.Keep original basis-column order and right-side roles rather than semantic sorting. Step identifiers and dependenciesRule: quotient_full_matrix_real_linear_equations_not_grid · Uses: G001, S012 S014 · calculationBefore quotient coset original difference membership pivot0 column0 normalized entry0 := div(1, 1) After quotient coset original difference membership pivot0 column0 normalized entry0 = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore quotient coset original difference membership pivot0 column0 normalized entry1 := div(-2, 1) After quotient coset original difference membership pivot0 column0 normalized entry1 = -2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore quotient coset original difference membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, -2] After normalized FULL row=[1, -2] Conditions: none Divide every entry by the same nonzero pivot,not only the left block or one entry. Step identifiers and dependenciesRule: quotient_nonzero_pivot_all_entry_scaling · Uses: G001, S015 S017 · calculationBefore quotient coset original difference membership eliminate row1 pivot0 entry0 product := mul(10, 1) After quotient coset original difference membership eliminate row1 pivot0 entry0 product = 10 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore quotient coset original difference membership eliminate row1 pivot0 entry0 difference := sub(10, 10) After quotient coset original difference membership eliminate row1 pivot0 entry0 difference = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore quotient coset original difference membership eliminate row1 pivot0 entry1 product := mul(10, -2) After quotient coset original difference membership eliminate row1 pivot0 entry1 product = -20 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore quotient coset original difference membership eliminate row1 pivot0 entry1 difference := sub(-20, -20) After quotient coset original difference membership eliminate row1 pivot0 entry1 difference = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore quotient coset original difference membership:target row=1;old row=[10, -20];subtract factor=10 times FULL pivot row=[1, -2] After new exact FULL row=[0, 0] Conditions: none Subtract the same multiple in every entry.The inverse adds it back,preserving every equation rather than replacing the full system with sampled solutions. Step identifiers and dependenciesRule: quotient_invertible_complete_row_elimination · Uses: G001, S020 S022 · impliesBefore quotient coset original difference membership:column=1;remaining exact entries=[0, 0] After no pivot added;current rank=1 Conditions: none Do not add a pivot for an all-zero remaining column;retain its original role and index. Step identifiers and dependenciesRule: quotient_zero_remaining_column_no_fake_pivot · Uses: G001, S021 S023 · impliesBefore quotient coset original difference membership:complete exact reduced matrix=[[1, -2], [0, 0], [0, 0]];ORIGINAL pivot-column indices=[0] After rank=1;pivot coordinates independent;ALL original column relations and full real system solution set preserved Conditions: none Pivot identity coordinates prove independence and express other columns by their pivot-row coordinates,so pivot count gives rank.Invertible operations carry these relations back to the original matrix. Step identifiers and dependenciesRule: quotient_full_rank_and_solution_set_certificate · Uses: G001, S022 S024 · calculationBefore quotient coset original difference membership original coordinate0 reconstruction coefficient0 product := mul(1, -2) After quotient coset original difference membership original coordinate0 reconstruction coefficient0 product = -2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore quotient coset original difference membership original coordinate0 reconstruction complete sum := sum(-2) After quotient coset original difference membership original coordinate0 reconstruction complete sum = -2 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore quotient coset original difference membership original coordinate1 reconstruction coefficient0 product := mul(10, -2) After quotient coset original difference membership original coordinate1 reconstruction coefficient0 product = -20 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore quotient coset original difference membership original coordinate1 reconstruction complete sum := sum(-20) After quotient coset original difference membership original coordinate1 reconstruction complete sum = -20 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore quotient coset original difference membership original coordinate2 reconstruction coefficient0 product := mul(0, -2) After quotient coset original difference membership original coordinate2 reconstruction coefficient0 product = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore quotient coset original difference membership original coordinate2 reconstruction complete sum := sum(0) After quotient coset original difference membership original coordinate2 reconstruction complete sum = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore quotient coset original difference membership:original proposed vector=[-2, -20, 0];original ordered independent columns=[[1, 10, 0]];complete reduced augmented system=[[1, -2], [0, 0], [0, 0]] After exact real-span coefficient witness=[-2];FULL original reconstruction=[-2, -20, 0];membership=Yes Conditions: none Read exact coefficients from the pivot equations and reconstruct every original coordinate.Empty coefficients reconstruct only the zero vector for an empty basis.Valid coefficients for nonempty bases may be fractional or outside original input bounds and must not be excluded. Step identifiers and dependenciesRule: quotient_full_span_coefficient_witness_not_sampled_membership · Uses: G001, S029 S031 · impliesBefore ORIGINAL v=[11, 2, 3],u=[13, 22, 3];ORDERED difference v-u=[-2, -20, 0];full-real-span membership=Yes After v+W=u+W iff v-u in W;coset equality=Yes;not a test of representative vector equality Conditions: none If the cosets agree,v=v+0 belongs to u+W,so v−u lies in W.Conversely,if v−u lies in W,then v+w=u+(v−u+w)belongs to u+W,and the negative difference proves the reverse inclusion.The difference need not be0;representative vectors need not agree. Step identifiers and dependenciesRule: quotient_coset_equality_both_inclusions_not_representative_equality · Uses: G001, S030 ## Reverse checks - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute quotient coset original difference membership original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Recompute quotient coset original difference membership eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute quotient coset original difference membership eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute quotient coset original difference membership eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute quotient coset original difference membership eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute quotient coset original difference membership pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute quotient coset original difference membership pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute quotient ORIGINAL v minus u coordinate2 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute quotient ORIGINAL v minus u coordinate1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute quotient ORIGINAL v minus u coordinate0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Recompute quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([-2, -20, 0], Yes, v_plus_W_equals_u_plus_W_iff_difference_in_W, quotient_elements_are_cosets_not_unique_representatives) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 7d4a4cc3b171f0041ff4d59a9ffa214966bf4ba53826f93a03b9e48543c18ae8 - candidate record SHA-256: ecd013161c6a2432d012148531f3b01c74314a5c60cf3b1ae673923a5502bce8 - semantic review receipt SHA-256: d165cad30849a3aba67390a59c572c0d965269d4b405e43df36da19e28d85612 - question-first witness SHA-256: c222ca47a07939c019865a1e4d9ce633205cdee68db92bf96a96f5683a62ebc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261106-r07-quotient-vector-space-0010.json)