Step-by-step derivation
S001 · implies
Before
source operation=quotient-coset-operations;original snapshot and mathematics retained
After
explicit ordered source output fields=[sum_representative, scalar_representative, addition_law, scalar_law, boundary]
Conditions: none
Keep the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not mathematical evidence,and No retains the operation's claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
quotient ORIGINAL ordered W basis:COMPLETE exact original ordered matrix=[[1], [3], [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 dependencies
Rule: quotient_full_matrix_real_linear_equations_not_grid · Uses: G001, S001
S003 · calculation
Before
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
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 dependencies
Rule: quotient_nonzero_pivot_all_entry_scaling · Uses: G001, S003
S005 · calculation
Before
quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product := mul(3, 1)
After
quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 product = 3
Conditions: none
Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
quotient ORIGINAL ordered W basis eliminate row1 pivot0 entry0 difference := sub(3, 3)
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 dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies
Before
quotient ORIGINAL ordered W basis:target row=1;old row=[3];subtract factor=3 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 dependencies
Rule: quotient_invertible_complete_row_elimination · Uses: G001, S006
S008 · implies
Before
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 dependencies
Rule: quotient_full_rank_and_solution_set_certificate · Uses: G001, S007
S009 · implies
Before
ACTUAL ambient real space R^3;complete ORIGINAL independent basis columns of W=[[1, 3, 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 dependencies
Rule: quotient_declared_subspace_full_real_span_zero_coset · Uses: G001, S008
S010 · implies
Before
ACTUAL subspace W full real span in R^3;original basis=[[1, 3, 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 dependencies
Rule: quotient_arbitrary_representatives_well_defined_both_operations · Uses: G001, S009
S011 · calculation
Before
quotient sum representative coordinate0 := sum(4, 3)
After
quotient sum representative coordinate0 = 7
Conditions: none
Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · calculation
Before
quotient sum representative coordinate1 := sum(2, 3)
After
quotient sum representative coordinate1 = 5
Conditions: none
Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · calculation
Before
quotient sum representative coordinate2 := sum(3, 4)
After
quotient sum representative coordinate2 = 7
Conditions: none
Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · calculation
Before
quotient scaled representative coordinate0 := mul(3, 4)
After
quotient scaled representative coordinate0 = 12
Conditions: none
Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · calculation
Before
quotient scaled representative coordinate1 := mul(3, 2)
After
quotient scaled representative coordinate1 = 6
Conditions: none
Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · calculation
Before
quotient scaled representative coordinate2 := mul(3, 3)
After
quotient scaled representative coordinate2 = 9
Conditions: none
Substitute original roles and established intermediates exactly,retaining signs,denominators,indices and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · implies
Before
ORIGINAL representatives v=[4, 2, 3],u=[3, 3, 4];original real scalar=3
After
reported representative vectors v+u=[7, 5, 7],lambda*v=[12, 6, 9];resulting objects are their cosets,not a selected generic unique vector name
Conditions: none
Keep the exact representative vectors in the first fields and interpret them through the quotient laws.No canonical representative selector is introduced;the zero quotient element remains the entire W.
Step identifiers and dependencies
Rule: quotient_operation_representatives_not_coset_object_identity · Uses: G001, S016