# Question 19960: Finite Quotient Dimension Subtracts Subspace Dimension [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-0085) 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. R0038-19960 · q-20261014-advance-20261106-r07-quotient-vector-space-0085 ## 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, certificate). - 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. Give (ambient_dimension, dimension_w, quotient_dimension, identity, 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-dimension-formula;original snapshot and mathematics retained After explicit ordered source output fields=[ambient_dimension, dimension_w, quotient_dimension, identity, 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], [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 · impliesBefore quotient basis extension standard coordinate0:COMPLETE exact original ordered matrix=[[1, 1], [10, 0], [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, S009 S011 · calculationBefore quotient basis extension standard coordinate0 pivot0 column0 normalized entry0 := div(1, 1) After quotient basis extension standard coordinate0 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, S010 S012 · calculationBefore quotient basis extension standard coordinate0 pivot0 column0 normalized entry1 := div(1, 1) After quotient basis extension standard coordinate0 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, S011 S013 · impliesBefore quotient basis extension standard coordinate0: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, S012 S014 · calculationBefore quotient basis extension standard coordinate0 eliminate row1 pivot0 entry0 product := mul(10, 1) After quotient basis extension standard coordinate0 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, S013 S015 · calculationBefore quotient basis extension standard coordinate0 eliminate row1 pivot0 entry0 difference := sub(10, 10) After quotient basis extension standard coordinate0 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, S014 S016 · calculationBefore quotient basis extension standard coordinate0 eliminate row1 pivot0 entry1 product := mul(10, 1) After quotient basis extension standard coordinate0 eliminate row1 pivot0 entry1 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, S015 S017 · calculationBefore quotient basis extension standard coordinate0 eliminate row1 pivot0 entry1 difference := sub(0, 10) After quotient basis extension standard coordinate0 eliminate row1 pivot0 entry1 difference = -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 · impliesBefore quotient basis extension standard coordinate0:target row=1;old row=[10, 0];subtract factor=10 times FULL pivot row=[1, 1] After new exact FULL row=[0, -10] 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, S017 S019 · calculationBefore quotient basis extension standard coordinate0 pivot1 column1 normalized entry0 := div(0, -10) After quotient basis extension standard coordinate0 pivot1 column1 normalized entry0 = 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, S018 S020 · calculationBefore quotient basis extension standard coordinate0 pivot1 column1 normalized entry1 := div(-10, -10) After quotient basis extension standard coordinate0 pivot1 column1 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, S019 S021 · impliesBefore quotient basis extension standard coordinate0:row=1;column=1;nonzero exact pivot=-10;old FULL row=[0, -10] After normalized FULL row=[0, 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, S020 S022 · calculationBefore quotient basis extension standard coordinate0 eliminate row0 pivot1 entry0 product := mul(1, 0) After quotient basis extension standard coordinate0 eliminate row0 pivot1 entry0 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 basis extension standard coordinate0 eliminate row0 pivot1 entry0 difference := sub(1, 0) After quotient basis extension standard coordinate0 eliminate row0 pivot1 entry0 difference = 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, S022 S024 · calculationBefore quotient basis extension standard coordinate0 eliminate row0 pivot1 entry1 product := mul(1, 1) After quotient basis extension standard coordinate0 eliminate row0 pivot1 entry1 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, S023 S025 · calculationBefore quotient basis extension standard coordinate0 eliminate row0 pivot1 entry1 difference := sub(1, 1) After quotient basis extension standard coordinate0 eliminate row0 pivot1 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, S024 S026 · impliesBefore quotient basis extension standard coordinate0:target row=0;old row=[1, 1];subtract factor=1 times FULL pivot row=[0, 1] After new exact FULL row=[1, 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, S025 S027 · impliesBefore quotient basis extension standard coordinate0:complete exact reduced matrix=[[1, 0], [0, 1], [0, 0]];ORIGINAL pivot-column indices=[0, 1] After rank=2;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, S026 S028 · impliesBefore quotient basis extension standard coordinate0:original proposed vector=[1, 0, 0];original independent columns=[[1, 10, 0]];FULL reduced augmented system=[[1, 0], [0, 1], [0, 0]] After contradiction row1:0*all real coefficients=1;NOT in the full real span Conditions: none The equivalent complete system contains0=1,so no real coefficients solve it.This is a global nonmembership certificate,not failure to find bounded integer coefficients. Step identifiers and dependenciesRule: quotient_actual_inconsistent_real_system_not_bounded_search_failure · Uses: G001, S027 S029 · impliesBefore next ORIGINAL standard-coordinate vector e_0=[1, 0, 0];in previous full span=No After append this independent new vector;current full ordered independent family=[[1, 10, 0], [1, 0, 0]] Conditions: none Skip a vector already in the span;otherwise append it independently.Previously tested standard vectors remain in the growing span,so processing the standard basis spans all real vectors.This is an exact finite basis construction,not inference from arbitrary samples. Step identifiers and dependenciesRule: quotient_exact_basis_extension_not_dimension_from_samples · Uses: G001, S028 S030 · impliesBefore quotient basis extension standard coordinate1:COMPLETE exact original ordered matrix=[[1, 1, 0], [10, 0, 1], [0, 0, 0]];rows=3;columns=3 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, S029 S031 · calculationBefore quotient basis extension standard coordinate1 pivot0 column0 normalized entry0 := div(1, 1) After quotient basis extension standard coordinate1 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, S030 S032 · calculationBefore quotient basis extension standard coordinate1 pivot0 column0 normalized entry1 := div(1, 1) After quotient basis extension standard coordinate1 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, S031 S033 · calculationBefore quotient basis extension standard coordinate1 pivot0 column0 normalized entry2 := div(0, 1) After quotient basis extension standard coordinate1 pivot0 column0 normalized entry2 = 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, S032 S034 · impliesBefore quotient basis extension standard coordinate1:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 1, 0] After normalized FULL row=[1, 1, 0] 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, S033 S035 · calculationBefore quotient basis extension standard coordinate1 eliminate row1 pivot0 entry0 product := mul(10, 1) After quotient basis extension standard coordinate1 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, S034 S036 · calculationBefore quotient basis extension standard coordinate1 eliminate row1 pivot0 entry0 difference := sub(10, 10) After quotient basis extension standard coordinate1 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, S035 S037 · calculationBefore quotient basis extension standard coordinate1 eliminate row1 pivot0 entry1 product := mul(10, 1) After quotient basis extension standard coordinate1 eliminate row1 pivot0 entry1 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, S036 S038 · calculationBefore quotient basis extension standard coordinate1 eliminate row1 pivot0 entry1 difference := sub(0, 10) After quotient basis extension standard coordinate1 eliminate row1 pivot0 entry1 difference = -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, S037 S039 · calculationBefore quotient basis extension standard coordinate1 eliminate row1 pivot0 entry2 product := mul(10, 0) After quotient basis extension standard coordinate1 eliminate row1 pivot0 entry2 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, S038 S040 · calculationBefore quotient basis extension standard coordinate1 eliminate row1 pivot0 entry2 difference := sub(1, 0) After quotient basis extension standard coordinate1 eliminate row1 pivot0 entry2 difference = 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, S039 S041 · impliesBefore quotient basis extension standard coordinate1:target row=1;old row=[10, 0, 1];subtract factor=10 times FULL pivot row=[1, 1, 0] After new exact FULL row=[0, -10, 1] 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, S040 S042 · calculationBefore quotient basis extension standard coordinate1 pivot1 column1 normalized entry0 := div(0, -10) After quotient basis extension standard coordinate1 pivot1 column1 normalized entry0 = 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, S041 S043 · calculationBefore quotient basis extension standard coordinate1 pivot1 column1 normalized entry1 := div(-10, -10) After quotient basis extension standard coordinate1 pivot1 column1 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, S042 S044 · calculationBefore quotient basis extension standard coordinate1 pivot1 column1 normalized entry2 := div(1, -10) After quotient basis extension standard coordinate1 pivot1 column1 normalized entry2 = -1/10 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, S043 S045 · impliesBefore quotient basis extension standard coordinate1:row=1;column=1;nonzero exact pivot=-10;old FULL row=[0, -10, 1] After normalized FULL row=[0, 1, -1/10] 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, S044 S046 · calculationBefore quotient basis extension standard coordinate1 eliminate row0 pivot1 entry0 product := mul(1, 0) After quotient basis extension standard coordinate1 eliminate row0 pivot1 entry0 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, S045 S047 · calculationBefore quotient basis extension standard coordinate1 eliminate row0 pivot1 entry0 difference := sub(1, 0) After quotient basis extension standard coordinate1 eliminate row0 pivot1 entry0 difference = 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, S046 S048 · calculationBefore quotient basis extension standard coordinate1 eliminate row0 pivot1 entry1 product := mul(1, 1) After quotient basis extension standard coordinate1 eliminate row0 pivot1 entry1 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, S047 S049 · calculationBefore quotient basis extension standard coordinate1 eliminate row0 pivot1 entry1 difference := sub(1, 1) After quotient basis extension standard coordinate1 eliminate row0 pivot1 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, S048 S050 · calculationBefore quotient basis extension standard coordinate1 eliminate row0 pivot1 entry2 product := mul(1, -1/10) After quotient basis extension standard coordinate1 eliminate row0 pivot1 entry2 product = -1/10 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, S049 S051 · calculationBefore quotient basis extension standard coordinate1 eliminate row0 pivot1 entry2 difference := sub(0, -1/10) After quotient basis extension standard coordinate1 eliminate row0 pivot1 entry2 difference = 1/10 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, S050 S052 · impliesBefore quotient basis extension standard coordinate1:target row=0;old row=[1, 1, 0];subtract factor=1 times FULL pivot row=[0, 1, -1/10] After new exact FULL row=[1, 0, 1/10] 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, S051 S053 · impliesBefore quotient basis extension standard coordinate1:column=2;remaining exact entries=[0] After no pivot added;current rank=2 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, S052 S054 · impliesBefore quotient basis extension standard coordinate1:complete exact reduced matrix=[[1, 0, 1/10], [0, 1, -1/10], [0, 0, 0]];ORIGINAL pivot-column indices=[0, 1] After rank=2;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, S053 S055 · calculationBefore quotient basis extension standard coordinate1 original coordinate0 reconstruction coefficient0 product := mul(1, 1/10) After quotient basis extension standard coordinate1 original coordinate0 reconstruction coefficient0 product = 1/10 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, S054 S056 · calculationBefore quotient basis extension standard coordinate1 original coordinate0 reconstruction coefficient1 product := mul(1, -1/10) After quotient basis extension standard coordinate1 original coordinate0 reconstruction coefficient1 product = -1/10 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, S055 S057 · calculationBefore quotient basis extension standard coordinate1 original coordinate0 reconstruction complete sum := sum(1/10, -1/10) After quotient basis extension standard coordinate1 original coordinate0 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, S056 S058 · calculationBefore quotient basis extension standard coordinate1 original coordinate1 reconstruction coefficient0 product := mul(10, 1/10) After quotient basis extension standard coordinate1 original coordinate1 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, S057 S059 · calculationBefore quotient basis extension standard coordinate1 original coordinate1 reconstruction coefficient1 product := mul(0, -1/10) After quotient basis extension standard coordinate1 original coordinate1 reconstruction coefficient1 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, S058 S060 · calculationBefore quotient basis extension standard coordinate1 original coordinate1 reconstruction complete sum := sum(1, 0) After quotient basis extension standard coordinate1 original coordinate1 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, S059 S061 · calculationBefore quotient basis extension standard coordinate1 original coordinate2 reconstruction coefficient0 product := mul(0, 1/10) After quotient basis extension standard coordinate1 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, S060 S062 · calculationBefore quotient basis extension standard coordinate1 original coordinate2 reconstruction coefficient1 product := mul(0, -1/10) After quotient basis extension standard coordinate1 original coordinate2 reconstruction coefficient1 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, S061 S063 · calculationBefore quotient basis extension standard coordinate1 original coordinate2 reconstruction complete sum := sum(0, 0) After quotient basis extension standard coordinate1 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, S062 S064 · impliesBefore quotient basis extension standard coordinate1:original proposed vector=[0, 1, 0];original ordered independent columns=[[1, 10, 0], [1, 0, 0]];complete reduced augmented system=[[1, 0, 1/10], [0, 1, -1/10], [0, 0, 0]] After exact real-span coefficient witness=[1/10, -1/10];FULL original reconstruction=[0, 1, 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, S063 S065 · impliesBefore next ORIGINAL standard-coordinate vector e_1=[0, 1, 0];in previous full span=Yes After keep previous basis;current full ordered independent family=[[1, 10, 0], [1, 0, 0]] Conditions: none Skip a vector already in the span;otherwise append it independently.Previously tested standard vectors remain in the growing span,so processing the standard basis spans all real vectors.This is an exact finite basis construction,not inference from arbitrary samples. Step identifiers and dependenciesRule: quotient_exact_basis_extension_not_dimension_from_samples · Uses: G001, S064 S066 · impliesBefore quotient basis extension standard coordinate2:COMPLETE exact original ordered matrix=[[1, 1, 0], [10, 0, 0], [0, 0, 1]];rows=3;columns=3 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, S065 S067 · calculationBefore quotient basis extension standard coordinate2 pivot0 column0 normalized entry0 := div(1, 1) After quotient basis extension standard coordinate2 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, S066 S068 · calculationBefore quotient basis extension standard coordinate2 pivot0 column0 normalized entry1 := div(1, 1) After quotient basis extension standard coordinate2 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, S067 S069 · calculationBefore quotient basis extension standard coordinate2 pivot0 column0 normalized entry2 := div(0, 1) After quotient basis extension standard coordinate2 pivot0 column0 normalized entry2 = 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, S068 S070 · impliesBefore quotient basis extension standard coordinate2:row=0;column=0;nonzero exact pivot=1;old FULL row=[1, 1, 0] After normalized FULL row=[1, 1, 0] 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 basis extension standard coordinate2 eliminate row1 pivot0 entry0 product := mul(10, 1) After quotient basis extension standard coordinate2 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, S070 S072 · calculationBefore quotient basis extension standard coordinate2 eliminate row1 pivot0 entry0 difference := sub(10, 10) After quotient basis extension standard coordinate2 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 basis extension standard coordinate2 eliminate row1 pivot0 entry1 product := mul(10, 1) After quotient basis extension standard coordinate2 eliminate row1 pivot0 entry1 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, S072 S074 · calculationBefore quotient basis extension standard coordinate2 eliminate row1 pivot0 entry1 difference := sub(0, 10) After quotient basis extension standard coordinate2 eliminate row1 pivot0 entry1 difference = -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, S073 S075 · calculationBefore quotient basis extension standard coordinate2 eliminate row1 pivot0 entry2 product := mul(10, 0) After quotient basis extension standard coordinate2 eliminate row1 pivot0 entry2 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, S074 S076 · calculationBefore quotient basis extension standard coordinate2 eliminate row1 pivot0 entry2 difference := sub(0, 0) After quotient basis extension standard coordinate2 eliminate row1 pivot0 entry2 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, S075 S077 · impliesBefore quotient basis extension standard coordinate2:target row=1;old row=[10, 0, 0];subtract factor=10 times FULL pivot row=[1, 1, 0] After new exact FULL row=[0, -10, 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, S076 S078 · calculationBefore quotient basis extension standard coordinate2 pivot1 column1 normalized entry0 := div(0, -10) After quotient basis extension standard coordinate2 pivot1 column1 normalized entry0 = 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, S077 S079 · calculationBefore quotient basis extension standard coordinate2 pivot1 column1 normalized entry1 := div(-10, -10) After quotient basis extension standard coordinate2 pivot1 column1 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, S078 S080 · calculationBefore quotient basis extension standard coordinate2 pivot1 column1 normalized entry2 := div(0, -10) After quotient basis extension standard coordinate2 pivot1 column1 normalized entry2 = 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, S079 S081 · impliesBefore quotient basis extension standard coordinate2:row=1;column=1;nonzero exact pivot=-10;old FULL row=[0, -10, 0] After normalized FULL row=[0, 1, 0] 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, S080 S082 · calculationBefore quotient basis extension standard coordinate2 eliminate row0 pivot1 entry0 product := mul(1, 0) After quotient basis extension standard coordinate2 eliminate row0 pivot1 entry0 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 basis extension standard coordinate2 eliminate row0 pivot1 entry0 difference := sub(1, 0) After quotient basis extension standard coordinate2 eliminate row0 pivot1 entry0 difference = 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, S082 S084 · calculationBefore quotient basis extension standard coordinate2 eliminate row0 pivot1 entry1 product := mul(1, 1) After quotient basis extension standard coordinate2 eliminate row0 pivot1 entry1 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, S083 S085 · calculationBefore quotient basis extension standard coordinate2 eliminate row0 pivot1 entry1 difference := sub(1, 1) After quotient basis extension standard coordinate2 eliminate row0 pivot1 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, S084 S086 · calculationBefore quotient basis extension standard coordinate2 eliminate row0 pivot1 entry2 product := mul(1, 0) After quotient basis extension standard coordinate2 eliminate row0 pivot1 entry2 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, S085 S087 · calculationBefore quotient basis extension standard coordinate2 eliminate row0 pivot1 entry2 difference := sub(0, 0) After quotient basis extension standard coordinate2 eliminate row0 pivot1 entry2 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, S086 S088 · impliesBefore quotient basis extension standard coordinate2:target row=0;old row=[1, 1, 0];subtract factor=1 times FULL pivot row=[0, 1, 0] After new exact FULL row=[1, 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, S087 S089 · calculationBefore quotient basis extension standard coordinate2 pivot2 column2 normalized entry0 := div(0, 1) After quotient basis extension standard coordinate2 pivot2 column2 normalized entry0 = 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, S088 S090 · calculationBefore quotient basis extension standard coordinate2 pivot2 column2 normalized entry1 := div(0, 1) After quotient basis extension standard coordinate2 pivot2 column2 normalized entry1 = 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 basis extension standard coordinate2 pivot2 column2 normalized entry2 := div(1, 1) After quotient basis extension standard coordinate2 pivot2 column2 normalized entry2 = 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, S090 S092 · impliesBefore quotient basis extension standard coordinate2:row=2;column=2;nonzero exact pivot=1;old FULL row=[0, 0, 1] After normalized FULL row=[0, 0, 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, S091 S093 · impliesBefore quotient basis extension standard coordinate2:complete exact reduced matrix=[[1, 0, 0], [0, 1, 0], [0, 0, 1]];ORIGINAL pivot-column indices=[0, 1, 2] After rank=3;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, S092 S094 · impliesBefore quotient basis extension standard coordinate2:original proposed vector=[0, 0, 1];original independent columns=[[1, 10, 0], [1, 0, 0]];FULL reduced augmented system=[[1, 0, 0], [0, 1, 0], [0, 0, 1]] After contradiction row2:0*all real coefficients=1;NOT in the full real span Conditions: none The equivalent complete system contains0=1,so no real coefficients solve it.This is a global nonmembership certificate,not failure to find bounded integer coefficients. Step identifiers and dependenciesRule: quotient_actual_inconsistent_real_system_not_bounded_search_failure · Uses: G001, S093 S095 · impliesBefore next ORIGINAL standard-coordinate vector e_2=[0, 0, 1];in previous full span=No After append this independent new vector;current full ordered independent family=[[1, 10, 0], [1, 0, 0], [0, 0, 1]] Conditions: none Skip a vector already in the span;otherwise append it independently.Previously tested standard vectors remain in the growing span,so processing the standard basis spans all real vectors.This is an exact finite basis construction,not inference from arbitrary samples. Step identifiers and dependenciesRule: quotient_exact_basis_extension_not_dimension_from_samples · Uses: G001, S094 S096 · calculationBefore quotient finite dimension difference := sub(3, 1) After quotient finite dimension difference = 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, S095 S097 · impliesBefore ORIGINAL complete W basis=[[1, 10, 0]];exact full ambient basis extends it by=[[1, 0, 0], [0, 0, 1]];ambient dimension=3 After cosets of extension vectors span V/W;any zero coset combination would be a full-basis relation,so all its coefficients0;dim(V/W)=2=n-dimW;zero coset=WHOLE W Conditions: none Expand any ambient vector in the full basis;its W part vanishes in the quotient,so the extension cosets span.If an extension combination belongs to W,it gives a zero relation in the full independent basis,forcing every extension coefficient0.Thus these cosets form a quotient basis,including W=0 and W=V.This is a finite-dimensional formula. Step identifiers and dependenciesRule: quotient_basis_extension_full_span_independence_dimension · Uses: G001, S096 ## Reverse checks - S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S096 · premise_and_result_audit — Recompute quotient finite dimension difference 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S091 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 pivot2 column2 normalized entry2 with exact rational arithmetic, not rounded decimals. - S090 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 pivot2 column2 normalized entry1 with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 pivot2 column2 normalized entry0 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 basis extension standard coordinate2 eliminate row0 pivot1 entry2 difference with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 eliminate row0 pivot1 entry2 product with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 eliminate row0 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 eliminate row0 pivot1 entry1 product with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 eliminate row0 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 eliminate row0 pivot1 entry0 product with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S080 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 pivot1 column1 normalized entry2 with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 pivot1 column1 normalized entry0 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 — Recompute quotient basis extension standard coordinate2 eliminate row1 pivot0 entry2 difference with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 eliminate row1 pivot0 entry2 product with exact rational arithmetic, not rounded decimals. - S074 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 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 basis extension standard coordinate2 pivot0 column0 normalized entry2 with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute quotient basis extension standard coordinate2 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S063 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 original coordinate2 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 original coordinate2 reconstruction coefficient1 product with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 original coordinate2 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 original coordinate1 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 original coordinate1 reconstruction coefficient1 product with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 original coordinate1 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 original coordinate0 reconstruction complete sum with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 original coordinate0 reconstruction coefficient1 product with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 original coordinate0 reconstruction coefficient0 product with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 basis extension standard coordinate1 eliminate row0 pivot1 entry2 difference with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 eliminate row0 pivot1 entry2 product with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 eliminate row0 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 eliminate row0 pivot1 entry1 product with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 eliminate row0 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 eliminate row0 pivot1 entry0 product with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 pivot1 column1 normalized entry2 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 eliminate row1 pivot0 entry2 difference with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 eliminate row1 pivot0 entry2 product with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 eliminate row1 pivot0 entry0 product with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 pivot0 column0 normalized entry2 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute quotient basis extension standard coordinate1 pivot0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Recompute quotient basis extension standard coordinate0 eliminate row0 pivot1 entry1 difference with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute quotient basis extension standard coordinate0 eliminate row0 pivot1 entry1 product with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute quotient basis extension standard coordinate0 eliminate row0 pivot1 entry0 difference with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute quotient basis extension standard coordinate0 eliminate row0 pivot1 entry0 product with exact rational arithmetic, not rounded decimals. - 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 basis extension standard coordinate0 pivot1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute quotient basis extension standard coordinate0 pivot1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute quotient basis extension standard coordinate0 eliminate row1 pivot0 entry1 difference with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute quotient basis extension standard coordinate0 eliminate row1 pivot0 entry1 product with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute quotient basis extension standard coordinate0 eliminate row1 pivot0 entry0 difference with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute quotient basis extension standard coordinate0 eliminate row1 pivot0 entry0 product 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 basis extension standard coordinate0 pivot0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute quotient basis extension standard coordinate0 pivot0 column0 normalized entry0 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 (3, 1, 2, Yes, finite_dimensional_dimension_formula_and_zero_coset_is_W) 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: 7259a6865e404a8ed31803f4baf6fec247ee989c313e24f23e86d28049621431 - candidate record SHA-256: ca7d33bc03603128ef64be669dd82fa93d5e3a559befa4bad613e5329ceedb89 - 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-0085.json)