# Question 20000: Changing Representatives by W Must Preserve Both Operations [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-0075) 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. R0038-20000 · q-20261014-advance-20261106-r07-quotient-vector-space-0075 ## Original question and givens 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). - G001 — 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). ## 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-representative-independence;original snapshot and mathematics retained After explicit ordered source output fields=[shift_v_in_w, shift_u_in_w, sum_difference_in_w, scalar_difference_in_w, well_defined_certificate, certificate] 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], [25], [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(25, 1) After quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product = 25 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(25, 25) 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=[25];subtract factor=25 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, 25, 0]] After W is the FULL real span of these columns;dim W=1;zero coset is the WHOLE subspace W,not the vector0 alone Conditions: none Exact rank verifies independence of the declared complete basis,whose count is dim W.The list is not the set of all W elements.An empty basis gives W={0};the zero coset is still the set W,not a representative vector. Step identifiers and dependenciesRule: quotient_declared_subspace_full_real_span_zero_coset · Uses: G001, S008 S010 · impliesBefore ACTUAL subspace W full real span in R^3;original basis=[[1, 25, 0]] After for ANY representatives v'=v+w,u'=u+z with w,z in W,and ANY real lambda,sum difference=w+z in W and scalar difference=lambda*w in W;thus quotient addition/scaling independent of representatives Conditions: none Subspace closure under addition,negation and every real scalar keeps all representative-change differences in W.This proves well-definedness for arbitrary changes,not just supplied shifts.Quotient objects are cosets:representatives are generally nonunique,with uniqueness possible when W={0},without equating the coset object type with vectors. Step identifiers and dependenciesRule: quotient_arbitrary_representatives_well_defined_both_operations · Uses: G001, S009 S011 · calculationBefore quotient original v shift original coordinate0 coefficient0 product := mul(1, 26) After quotient original v shift original coordinate0 coefficient0 product = 26 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 shift original coordinate0 complete sum := sum(26) After quotient original v shift original coordinate0 complete sum = 26 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore quotient original v shift original coordinate1 coefficient0 product := mul(25, 26) After quotient original v shift original coordinate1 coefficient0 product = 650 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore quotient original v shift original coordinate1 complete sum := sum(650) After quotient original v shift original coordinate1 complete sum = 650 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 original v shift original coordinate2 coefficient0 product := mul(0, 26) After quotient original v shift original coordinate2 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, S014 S016 · calculationBefore quotient original v shift original coordinate2 complete sum := sum(0) After quotient original v shift original coordinate2 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, S015 S017 · impliesBefore quotient original v shift:ORIGINAL W basis order=[[1, 25, 0]];aligned coefficient block=[26] After computed whole vector=[26, 650, 0];a real linear combination in W by its declared span Conditions: none Align coefficients with the original basis columns,without semantic reordering.The complete linear combination itself witnesses membership in W. Step identifiers and dependenciesRule: quotient_original_basis_coefficients_not_semantic_reordering · Uses: G001, S016 S018 · calculationBefore quotient original u shift original coordinate0 coefficient0 product := mul(1, -25) After quotient original u shift original coordinate0 coefficient0 product = -25 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 original u shift original coordinate0 complete sum := sum(-25) After quotient original u shift original coordinate0 complete sum = -25 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 original u shift original coordinate1 coefficient0 product := mul(25, -25) After quotient original u shift original coordinate1 coefficient0 product = -625 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 · calculationBefore quotient original u shift original coordinate1 complete sum := sum(-625) After quotient original u shift original coordinate1 complete sum = -625 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, S020 S022 · calculationBefore quotient original u shift original coordinate2 coefficient0 product := mul(0, -25) After quotient original u shift original coordinate2 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, S021 S023 · calculationBefore quotient original u shift original coordinate2 complete sum := sum(0) After quotient original u shift original coordinate2 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, S022 S024 · impliesBefore quotient original u shift:ORIGINAL W basis order=[[1, 25, 0]];aligned coefficient block=[-25] After computed whole vector=[-25, -625, 0];a real linear combination in W by its declared span Conditions: none Align coefficients with the original basis columns,without semantic reordering.The complete linear combination itself witnesses membership in W. Step identifiers and dependenciesRule: quotient_original_basis_coefficients_not_semantic_reordering · Uses: G001, S023 S025 · calculationBefore quotient changed v coordinate0 := sum(26, 26) After quotient changed v coordinate0 = 52 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 changed v coordinate1 := sum(2, 650) After quotient changed v coordinate1 = 652 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 changed v coordinate2 := sum(3, 0) After quotient changed v coordinate2 = 3 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 changed u coordinate0 := sum(3, -25) After quotient changed u coordinate0 = -22 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 changed u coordinate1 := sum(25, -625) After quotient changed u coordinate1 = -600 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 · calculationBefore quotient changed u coordinate2 := sum(4, 0) After quotient changed u coordinate2 = 4 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, S029 S031 · calculationBefore quotient new sum coordinate0 := sum(52, -22) After quotient new sum coordinate0 = 30 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, S030 S032 · calculationBefore quotient old sum coordinate0 := sum(26, 3) After quotient old sum coordinate0 = 29 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, S031 S033 · calculationBefore quotient changed sum difference coordinate0 := sub(30, 29) After quotient changed sum difference coordinate0 = 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, S032 S034 · calculationBefore quotient new scalar representative coordinate0 := mul(25, 52) After quotient new scalar representative coordinate0 = 1300 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, S033 S035 · calculationBefore quotient old scalar representative coordinate0 := mul(25, 26) After quotient old scalar representative coordinate0 = 650 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, S034 S036 · calculationBefore quotient changed scalar difference coordinate0 := sub(1300, 650) After quotient changed scalar difference coordinate0 = 650 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, S035 S037 · calculationBefore quotient new sum coordinate1 := sum(652, -600) After quotient new sum coordinate1 = 52 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, S036 S038 · calculationBefore quotient old sum coordinate1 := sum(2, 25) After quotient old sum coordinate1 = 27 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, S037 S039 · calculationBefore quotient changed sum difference coordinate1 := sub(52, 27) After quotient changed sum difference coordinate1 = 25 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, S038 S040 · calculationBefore quotient new scalar representative coordinate1 := mul(25, 652) After quotient new scalar representative coordinate1 = 16300 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, S039 S041 · calculationBefore quotient old scalar representative coordinate1 := mul(25, 2) After quotient old scalar representative coordinate1 = 50 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, S040 S042 · calculationBefore quotient changed scalar difference coordinate1 := sub(16300, 50) After quotient changed scalar difference coordinate1 = 16250 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, S041 S043 · calculationBefore quotient new sum coordinate2 := sum(3, 4) After quotient new sum coordinate2 = 7 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore quotient old sum coordinate2 := sum(3, 4) After quotient old sum coordinate2 = 7 Conditions: none Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore quotient changed sum difference coordinate2 := sub(7, 7) After quotient changed sum difference 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, S044 S046 · calculationBefore quotient new scalar representative coordinate2 := mul(25, 3) After quotient new scalar representative coordinate2 = 75 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, S045 S047 · calculationBefore quotient old scalar representative coordinate2 := mul(25, 3) After quotient old scalar representative coordinate2 = 75 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, S046 S048 · calculationBefore quotient changed scalar difference coordinate2 := sub(75, 75) After quotient changed scalar difference 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, S047 S049 · impliesBefore quotient supplied v shift membership:COMPLETE exact original ordered matrix=[[1, 26], [25, 650], [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, S048 S050 · calculationBefore quotient supplied v shift membership pivot0 column0 normalized entry0 := div(1, 1) After quotient supplied v shift 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, S049 S051 · calculationBefore quotient supplied v shift membership pivot0 column0 normalized entry1 := div(26, 1) After quotient supplied v shift membership pivot0 column0 normalized entry1 = 26 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, S050 S052 · impliesBefore quotient supplied v shift membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 26] After normalized FULL row=[1, 26] 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, S051 S053 · calculationBefore quotient supplied v shift membership eliminate row1 pivot0 entry0 product := mul(25, 1) After quotient supplied v shift membership eliminate row1 pivot0 entry0 product = 25 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, S052 S054 · calculationBefore quotient supplied v shift membership eliminate row1 pivot0 entry0 difference := sub(25, 25) After quotient supplied v shift 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, S053 S055 · calculationBefore quotient supplied v shift membership eliminate row1 pivot0 entry1 product := mul(25, 26) After quotient supplied v shift membership eliminate row1 pivot0 entry1 product = 650 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, S054 S056 · calculationBefore quotient supplied v shift membership eliminate row1 pivot0 entry1 difference := sub(650, 650) After quotient supplied v shift 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, S055 S057 · impliesBefore quotient supplied v shift membership:target row=1;old row=[25, 650];subtract factor=25 times FULL pivot row=[1, 26] 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, S056 S058 · impliesBefore quotient supplied v shift 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, S057 S059 · impliesBefore quotient supplied v shift membership:complete exact reduced matrix=[[1, 26], [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, S058 S060 · calculationBefore quotient supplied v shift membership original coordinate0 reconstruction coefficient0 product := mul(1, 26) After quotient supplied v shift membership original coordinate0 reconstruction coefficient0 product = 26 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, S059 S061 · calculationBefore quotient supplied v shift membership original coordinate0 reconstruction complete sum := sum(26) After quotient supplied v shift membership original coordinate0 reconstruction complete sum = 26 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, S060 S062 · calculationBefore quotient supplied v shift membership original coordinate1 reconstruction coefficient0 product := mul(25, 26) After quotient supplied v shift membership original coordinate1 reconstruction coefficient0 product = 650 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, S061 S063 · calculationBefore quotient supplied v shift membership original coordinate1 reconstruction complete sum := sum(650) After quotient supplied v shift membership original coordinate1 reconstruction complete sum = 650 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, S062 S064 · calculationBefore quotient supplied v shift membership original coordinate2 reconstruction coefficient0 product := mul(0, 26) After quotient supplied v shift 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, S063 S065 · calculationBefore quotient supplied v shift membership original coordinate2 reconstruction complete sum := sum(0) After quotient supplied v shift 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, S064 S066 · impliesBefore quotient supplied v shift membership:original proposed vector=[26, 650, 0];original ordered independent columns=[[1, 25, 0]];complete reduced augmented system=[[1, 26], [0, 0], [0, 0]] After exact real-span coefficient witness=[26];FULL original reconstruction=[26, 650, 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, S065 S067 · impliesBefore quotient supplied u shift membership:COMPLETE exact original ordered matrix=[[1, -25], [25, -625], [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, S066 S068 · calculationBefore quotient supplied u shift membership pivot0 column0 normalized entry0 := div(1, 1) After quotient supplied u shift 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, S067 S069 · calculationBefore quotient supplied u shift membership pivot0 column0 normalized entry1 := div(-25, 1) After quotient supplied u shift membership pivot0 column0 normalized entry1 = -25 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, S068 S070 · impliesBefore quotient supplied u shift membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, -25] After normalized FULL row=[1, -25] 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, S069 S071 · calculationBefore quotient supplied u shift membership eliminate row1 pivot0 entry0 product := mul(25, 1) After quotient supplied u shift membership eliminate row1 pivot0 entry0 product = 25 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, S070 S072 · calculationBefore quotient supplied u shift membership eliminate row1 pivot0 entry0 difference := sub(25, 25) After quotient supplied u shift 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, S071 S073 · calculationBefore quotient supplied u shift membership eliminate row1 pivot0 entry1 product := mul(25, -25) After quotient supplied u shift membership eliminate row1 pivot0 entry1 product = -625 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, S072 S074 · calculationBefore quotient supplied u shift membership eliminate row1 pivot0 entry1 difference := sub(-625, -625) After quotient supplied u shift 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, S073 S075 · impliesBefore quotient supplied u shift membership:target row=1;old row=[25, -625];subtract factor=25 times FULL pivot row=[1, -25] 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, S074 S076 · impliesBefore quotient supplied u shift 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, S075 S077 · impliesBefore quotient supplied u shift membership:complete exact reduced matrix=[[1, -25], [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, S076 S078 · calculationBefore quotient supplied u shift membership original coordinate0 reconstruction coefficient0 product := mul(1, -25) After quotient supplied u shift membership original coordinate0 reconstruction coefficient0 product = -25 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, S077 S079 · calculationBefore quotient supplied u shift membership original coordinate0 reconstruction complete sum := sum(-25) After quotient supplied u shift membership original coordinate0 reconstruction complete sum = -25 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, S078 S080 · calculationBefore quotient supplied u shift membership original coordinate1 reconstruction coefficient0 product := mul(25, -25) After quotient supplied u shift membership original coordinate1 reconstruction coefficient0 product = -625 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, S079 S081 · calculationBefore quotient supplied u shift membership original coordinate1 reconstruction complete sum := sum(-625) After quotient supplied u shift membership original coordinate1 reconstruction complete sum = -625 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, S080 S082 · calculationBefore quotient supplied u shift membership original coordinate2 reconstruction coefficient0 product := mul(0, -25) After quotient supplied u shift 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, S081 S083 · calculationBefore quotient supplied u shift membership original coordinate2 reconstruction complete sum := sum(0) After quotient supplied u shift 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, S082 S084 · impliesBefore quotient supplied u shift membership:original proposed vector=[-25, -625, 0];original ordered independent columns=[[1, 25, 0]];complete reduced augmented system=[[1, -25], [0, 0], [0, 0]] After exact real-span coefficient witness=[-25];FULL original reconstruction=[-25, -625, 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, S083 S085 · impliesBefore quotient supplied sum difference membership:COMPLETE exact original ordered matrix=[[1, 1], [25, 25], [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, S084 S086 · calculationBefore quotient supplied sum difference membership pivot0 column0 normalized entry0 := div(1, 1) After quotient supplied sum 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, S085 S087 · calculationBefore quotient supplied sum difference membership pivot0 column0 normalized entry1 := div(1, 1) After quotient supplied sum difference membership pivot0 column0 normalized entry1 = 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, S086 S088 · impliesBefore quotient supplied sum difference membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 1] After normalized FULL row=[1, 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, S087 S089 · calculationBefore quotient supplied sum difference membership eliminate row1 pivot0 entry0 product := mul(25, 1) After quotient supplied sum difference membership eliminate row1 pivot0 entry0 product = 25 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, S088 S090 · calculationBefore quotient supplied sum difference membership eliminate row1 pivot0 entry0 difference := sub(25, 25) After quotient supplied sum 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, S089 S091 · calculationBefore quotient supplied sum difference membership eliminate row1 pivot0 entry1 product := mul(25, 1) After quotient supplied sum difference membership eliminate row1 pivot0 entry1 product = 25 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, S090 S092 · calculationBefore quotient supplied sum difference membership eliminate row1 pivot0 entry1 difference := sub(25, 25) After quotient supplied sum 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, S091 S093 · impliesBefore quotient supplied sum difference membership:target row=1;old row=[25, 25];subtract factor=25 times FULL pivot row=[1, 1] 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, S092 S094 · impliesBefore quotient supplied sum 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, S093 S095 · impliesBefore quotient supplied sum difference membership:complete exact reduced matrix=[[1, 1], [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, S094 S096 · calculationBefore quotient supplied sum difference membership original coordinate0 reconstruction coefficient0 product := mul(1, 1) After quotient supplied sum difference membership original coordinate0 reconstruction coefficient0 product = 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, S095 S097 · calculationBefore quotient supplied sum difference membership original coordinate0 reconstruction complete sum := sum(1) After quotient supplied sum difference membership original coordinate0 reconstruction complete sum = 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, S096 S098 · calculationBefore quotient supplied sum difference membership original coordinate1 reconstruction coefficient0 product := mul(25, 1) After quotient supplied sum difference membership original coordinate1 reconstruction coefficient0 product = 25 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, S097 S099 · calculationBefore quotient supplied sum difference membership original coordinate1 reconstruction complete sum := sum(25) After quotient supplied sum difference membership original coordinate1 reconstruction complete sum = 25 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, S098 S100 · calculationBefore quotient supplied sum difference membership original coordinate2 reconstruction coefficient0 product := mul(0, 1) After quotient supplied sum 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, S099 S101 · calculationBefore quotient supplied sum difference membership original coordinate2 reconstruction complete sum := sum(0) After quotient supplied sum 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, S100 S102 · impliesBefore quotient supplied sum difference membership:original proposed vector=[1, 25, 0];original ordered independent columns=[[1, 25, 0]];complete reduced augmented system=[[1, 1], [0, 0], [0, 0]] After exact real-span coefficient witness=[1];FULL original reconstruction=[1, 25, 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, S101 S103 · impliesBefore quotient supplied scalar difference membership:COMPLETE exact original ordered matrix=[[1, 650], [25, 16250], [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, S102 S104 · calculationBefore quotient supplied scalar difference membership pivot0 column0 normalized entry0 := div(1, 1) After quotient supplied scalar 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, S103 S105 · calculationBefore quotient supplied scalar difference membership pivot0 column0 normalized entry1 := div(650, 1) After quotient supplied scalar difference membership pivot0 column0 normalized entry1 = 650 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, S104 S106 · impliesBefore quotient supplied scalar difference membership:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 650] After normalized FULL row=[1, 650] 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, S105 S107 · calculationBefore quotient supplied scalar difference membership eliminate row1 pivot0 entry0 product := mul(25, 1) After quotient supplied scalar difference membership eliminate row1 pivot0 entry0 product = 25 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, S106 S108 · calculationBefore quotient supplied scalar difference membership eliminate row1 pivot0 entry0 difference := sub(25, 25) After quotient supplied scalar 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, S107 S109 · calculationBefore quotient supplied scalar difference membership eliminate row1 pivot0 entry1 product := mul(25, 650) After quotient supplied scalar difference membership eliminate row1 pivot0 entry1 product = 16250 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, S108 S110 · calculationBefore quotient supplied scalar difference membership eliminate row1 pivot0 entry1 difference := sub(16250, 16250) After quotient supplied scalar 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, S109 S111 · impliesBefore quotient supplied scalar difference membership:target row=1;old row=[25, 16250];subtract factor=25 times FULL pivot row=[1, 650] 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, S110 S112 · impliesBefore quotient supplied scalar 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, S111 S113 · impliesBefore quotient supplied scalar difference membership:complete exact reduced matrix=[[1, 650], [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, S112 S114 · calculationBefore quotient supplied scalar difference membership original coordinate0 reconstruction coefficient0 product := mul(1, 650) After quotient supplied scalar difference membership original coordinate0 reconstruction coefficient0 product = 650 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, S113 S115 · calculationBefore quotient supplied scalar difference membership original coordinate0 reconstruction complete sum := sum(650) After quotient supplied scalar difference membership original coordinate0 reconstruction complete sum = 650 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, S114 S116 · calculationBefore quotient supplied scalar difference membership original coordinate1 reconstruction coefficient0 product := mul(25, 650) After quotient supplied scalar difference membership original coordinate1 reconstruction coefficient0 product = 16250 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, S115 S117 · calculationBefore quotient supplied scalar difference membership original coordinate1 reconstruction complete sum := sum(16250) After quotient supplied scalar difference membership original coordinate1 reconstruction complete sum = 16250 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, S116 S118 · calculationBefore quotient supplied scalar difference membership original coordinate2 reconstruction coefficient0 product := mul(0, 650) After quotient supplied scalar 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, S117 S119 · calculationBefore quotient supplied scalar difference membership original coordinate2 reconstruction complete sum := sum(0) After quotient supplied scalar 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, S118 S120 · impliesBefore quotient supplied scalar difference membership:original proposed vector=[650, 16250, 0];original ordered independent columns=[[1, 25, 0]];complete reduced augmented system=[[1, 650], [0, 0], [0, 0]] After exact real-span coefficient witness=[650];FULL original reconstruction=[650, 16250, 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, S119 S121 · impliesBefore ACTUAL original W shifts=[[26, 650, 0], [-25, -625, 0]];changed representatives=[[52, 652, 3], [-22, -600, 4]];actual sum/scalar differences=[[1, 25, 0], [650, 16250, 0]] After ALL four exact full-span memberships=[Yes, Yes, Yes, Yes];differences equal shift_v+shift_u and lambda*shift_v,matching the arbitrary-representative proof Conditions: none Check every supplied difference in the complete original W and connect it to the arbitrary-change algebra above.The four passing instances are not the sole universal argument.Keep legitimate lambda=0 and W=0 cases. Step identifiers and dependenciesRule: quotient_supplied_changes_match_general_well_definedness · Uses: G001, S120 ## Reverse checks - S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S119 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S118 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S116 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S115 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S114 · premise_and_result_audit — Recompute quotient supplied scalar difference membership original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S110 · premise_and_result_audit — Recompute quotient supplied scalar difference membership eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Recompute quotient supplied scalar difference membership eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals. - S108 · premise_and_result_audit — Recompute quotient supplied scalar difference membership eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S107 · premise_and_result_audit — Recompute quotient supplied scalar difference membership eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals. - S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S105 · premise_and_result_audit — Recompute quotient supplied scalar difference membership pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S104 · premise_and_result_audit — Recompute quotient supplied scalar difference membership pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S101 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S100 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S098 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute quotient supplied sum difference membership original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S092 · premise_and_result_audit — Recompute quotient supplied sum difference membership eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Recompute quotient supplied sum difference membership eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals. - S090 · premise_and_result_audit — Recompute quotient supplied sum difference membership eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute quotient supplied sum difference membership eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals. - S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S087 · premise_and_result_audit — Recompute quotient supplied sum difference membership pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Recompute quotient supplied sum difference membership pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S083 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute quotient supplied u shift membership original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S074 · premise_and_result_audit — Recompute quotient supplied u shift membership eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute quotient supplied u shift membership eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute quotient supplied u shift membership eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute quotient supplied u shift membership eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S069 · premise_and_result_audit — Recompute quotient supplied u shift membership pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute quotient supplied u shift membership pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S065 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute quotient supplied v shift membership original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S056 · premise_and_result_audit — Recompute quotient supplied v shift membership eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute quotient supplied v shift membership eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute quotient supplied v shift membership eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute quotient supplied v shift membership eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S051 · premise_and_result_audit — Recompute quotient supplied v shift membership pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute quotient supplied v shift membership pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S048 · premise_and_result_audit — Recompute quotient changed scalar difference coordinate2 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute quotient old scalar representative coordinate2 with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute quotient new scalar representative coordinate2 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute quotient changed sum difference coordinate2 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute quotient old sum coordinate2 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute quotient new sum coordinate2 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute quotient changed scalar difference coordinate1 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute quotient old scalar representative coordinate1 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute quotient new scalar representative coordinate1 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute quotient changed sum difference coordinate1 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute quotient old sum coordinate1 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute quotient new sum coordinate1 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute quotient changed scalar difference coordinate0 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute quotient old scalar representative coordinate0 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute quotient new scalar representative coordinate0 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute quotient changed sum difference coordinate0 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute quotient old sum coordinate0 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute quotient new sum coordinate0 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute quotient changed u coordinate2 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute quotient changed u coordinate1 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute quotient changed u coordinate0 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute quotient changed v coordinate2 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute quotient changed v coordinate1 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute quotient changed v coordinate0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute quotient original u shift original coordinate2 complete sum with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute quotient original u shift original coordinate2 coefficient0 product with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute quotient original u shift original coordinate1 complete sum with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute quotient original u shift original coordinate1 coefficient0 product with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute quotient original u shift original coordinate0 complete sum with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute quotient original u shift original coordinate0 coefficient0 product with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Recompute quotient original v shift original coordinate2 complete sum with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute quotient original v shift original coordinate2 coefficient0 product with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute quotient original v shift original coordinate1 complete sum with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute quotient original v shift original coordinate1 coefficient0 product with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute quotient original v shift original coordinate0 complete sum with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute quotient original v shift original coordinate0 coefficient0 product with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Recompute quotient ORIGINAL ordered W basis pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (Yes, Yes, Yes, Yes, Yes, addition_and_scalar_multiplication_independent_of_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: 832efabcd18a82cc3fba30c7ccf084d3b0f4448106e05dcee07a4386b83c8598 - candidate record SHA-256: 138a2283497ac75c5553246719dba06dbec20267373f61ff86d3af866068ee87 - 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-0075.json)