Step-by-step derivation
S001 · implies
Before
same usual real ambient R^2;original ORDERED vectors as columns=[[1, 0], [6, 1]]
After
complete row-coordinate matrix=[[1, 6], [0, 1]];vector count=2;zero/repeated vectors retained
Conditions: none
Columns are the original vectors and rows their coordinates.Zero and repeated vectors are allowed,so semantic-template deduplication cannot replace the original family or its count.Exact rational row operations are invertible coordinate changes over the reals.
Step identifiers and dependencies
Rule: dimension_actual_columns_not_semantic_deduplication · Uses: G001
S002 · calculation
Before
dimension pivot row0 column0 normalized entry0 := div(1, 1)
After
dimension pivot row0 column0 normalized entry0 = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
dimension pivot row0 column0 normalized entry1 := div(6, 1)
After
dimension pivot row0 column0 normalized entry1 = 6
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
pivot row=0;pivot column=0;nonzero exact pivot=1;old row=[1, 6]
After
divide whole row by pivot;normalized row=[1, 6]
Conditions: none
Divide the whole row by the same nonzero pivot,showing every exact fraction.Do not change only the pivot cell or use floating-point approximations.Multiplying back by the pivot reverses the step.
Step identifiers and dependencies
Rule: dimension_nonzero_pivot_scaling_every_entry · Uses: G001, S003
S005 · calculation
Before
dimension pivot row1 column1 normalized entry0 := div(0, 1)
After
dimension pivot row1 column1 normalized entry0 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
dimension pivot row1 column1 normalized entry1 := div(1, 1)
After
dimension pivot row1 column1 normalized entry1 = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies
Before
pivot row=1;pivot column=1;nonzero exact pivot=1;old row=[0, 1]
After
divide whole row by pivot;normalized row=[0, 1]
Conditions: none
Divide the whole row by the same nonzero pivot,showing every exact fraction.Do not change only the pivot cell or use floating-point approximations.Multiplying back by the pivot reverses the step.
Step identifiers and dependencies
Rule: dimension_nonzero_pivot_scaling_every_entry · Uses: G001, S006
S008 · calculation
Before
dimension eliminate row0 pivot1 column1 entry0 multiple := mul(6, 0)
After
dimension eliminate row0 pivot1 column1 entry0 multiple = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
dimension eliminate row0 pivot1 column1 entry0 difference := sub(1, 0)
After
dimension eliminate row0 pivot1 column1 entry0 difference = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
dimension eliminate row0 pivot1 column1 entry1 multiple := mul(6, 1)
After
dimension eliminate row0 pivot1 column1 entry1 multiple = 6
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · calculation
Before
dimension eliminate row0 pivot1 column1 entry1 difference := sub(6, 6)
After
dimension eliminate row0 pivot1 column1 entry1 difference = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
target row=0;old row=[1, 6];subtract factor=6 times normalized pivot row=[0, 1]
After
new target row=[1, 0];entry at pivot column1=0
Conditions: none
Subtract the factor times the pivot row in every coordinate to clear this pivot column.Adding it back reverses the step,so column-coefficient equations remain equivalent.
Step identifiers and dependencies
Rule: dimension_exact_row_subtraction_all_coordinates · Uses: G001, S011
S013 · implies
Before
original column family count=2;ambient dimension=2;final exact matrix=[[1, 0], [0, 1]];pivot columns=[0, 1]
After
rank=2;independent=Yes;spans whole ambient=Yes;basis=Yes
Conditions: none
Invertible row operations preserve all real solutions of M*c=0.The pivot count is real rank,and independence requires rank equal to column count.Left multiplication by invertible E maps the original column span to its E-image,preserving dimension and whole-ambient spanning status,not literal equality of the two spans.Spanning requires rank=n;both conditions are needed for a basis.
Step identifiers and dependencies
Rule: dimension_pivots_relations_and_whole_ambient_spanning · Uses: G001, S012