Original question and givens
Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [7, 1]] in ambient R^2, limited to dimensions 2 or 3. Treat vectors as columns and calculate rank by exact rational Gaussian elimination, not floating-point samples. A basis is both independent and spanning. An independent family gives only a lower bound; a spanning family gives only an upper bound. Basis-cardinality invariance uses the finite-dimensional theorem, not a few examples. Give (vector_count,rank,spanning?,upper_bound). The last field is dimension_at_most_vector_count with the numeric count, or no_upper_bound_from_nonspanning_family.
G001 — Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [7, 1]] in ambient R^2, limited to dimensions 2 or 3. Treat vectors as columns and calculate rank by exact rational Gaussian elimination, not floating-point samples. A basis is both independent and spanning. An independent family gives only a lower bound; a spanning family gives only an upper bound. Basis-cardinality invariance uses the finite-dimensional theorem, not a few examples. Give (vector_count,rank,spanning?,upper_bound). The last field is dimension_at_most_vector_count with the numeric count, or no_upper_bound_from_nonspanning_family.
Step-by-step derivation
S001 · implies
Before
same usual real ambient R^2;original ORDERED vectors as columns=[[1, 0], [0, 1], [7, 1]]
After
complete row-coordinate matrix=[[1, 0, 7], [0, 1, 1]];vector count=3;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(0, 1)
After
dimension pivot row0 column0 normalized entry1 = 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, S002
S004 · calculation
Before
dimension pivot row0 column0 normalized entry2 := div(7, 1)
After
dimension pivot row0 column0 normalized entry2 = 7
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, S003
S005 · implies
Before
pivot row=0;pivot column=0;nonzero exact pivot=1;old row=[1, 0, 7]
After
divide whole row by pivot;normalized row=[1, 0, 7]
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, S004
S006 · 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, S005
S007 · 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, S006
S008 · calculation
Before
dimension pivot row1 column1 normalized entry2 := div(1, 1)
After
dimension pivot row1 column1 normalized entry2 = 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, S007
S009 · implies
Before
pivot row=1;pivot column=1;nonzero exact pivot=1;old row=[0, 1, 1]
After
divide whole row by pivot;normalized row=[0, 1, 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, S008
S010 · implies
Before
original column family count=3;ambient dimension=2;final exact matrix=[[1, 0, 7], [0, 1, 1]];pivot columns=[0, 1]
After
rank=2;independent=No;spans whole ambient=Yes;basis=No
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, S009
S011 · implies
Before
supplied original vector count=3;exact rank=2;whole-ambient spanning=Yes
After
this family's upper-bound certificate=dimension_at_most_3
Conditions: none
A finite spanning family can be pruned to a basis,so dimension≤m;the bound can be loose.If it does not span,this family supplies no such certificate,not proof of infinite ambient dimension.Do not silently replace the original m by a tighter answer.
Step identifiers and dependencies
Rule: dimension_spanning_family_only_its_possibly_loose_upper_bound · Uses: G001, S010