Question 16965: An Independent Family Gives Only a Lower Bound

Original question and givens

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [14, 1, 0]] in ambient R^3, 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,independent?,lower_bound). The last field is dimension_at_least_vector_count with the numeric count, or no_lower_bound_from_dependent_family.

• G001 — Use the complete ordered real coordinate family vectors=[[1, 0, 0], [14, 1, 0]] in ambient R^3, 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,independent?,lower_bound). The last field is dimension_at_least_vector_count with the numeric count, or no_lower_bound_from_dependent_family.

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-vector-space-dimension

Step-by-step derivation

same usual real ambient R^3;original ORDERED vectors as columns=[[1, 0, 0], [14, 1, 0]]

complete row-coordinate matrix=[[1, 14], [0, 1], [0, 0]];vector count=2;zero/repeated vectors retained

dimension pivot row0 column0 normalized entry0 := div(1, 1)

dimension pivot row0 column0 normalized entry0 = 1

dimension pivot row0 column0 normalized entry1 := div(14, 1)

dimension pivot row0 column0 normalized entry1 = 14

pivot row=0;pivot column=0;nonzero exact pivot=1;old row=[1, 14]

divide whole row by pivot;normalized row=[1, 14]

dimension pivot row1 column1 normalized entry0 := div(0, 1)

dimension pivot row1 column1 normalized entry0 = 0

dimension pivot row1 column1 normalized entry1 := div(1, 1)

dimension pivot row1 column1 normalized entry1 = 1

pivot row=1;pivot column=1;nonzero exact pivot=1;old row=[0, 1]

divide whole row by pivot;normalized row=[0, 1]

dimension eliminate row0 pivot1 column1 entry0 multiple := mul(14, 0)

dimension eliminate row0 pivot1 column1 entry0 multiple = 0

dimension eliminate row0 pivot1 column1 entry0 difference := sub(1, 0)

dimension eliminate row0 pivot1 column1 entry0 difference = 1

dimension eliminate row0 pivot1 column1 entry1 multiple := mul(14, 1)

dimension eliminate row0 pivot1 column1 entry1 multiple = 14

dimension eliminate row0 pivot1 column1 entry1 difference := sub(14, 14)

dimension eliminate row0 pivot1 column1 entry1 difference = 0

target row=0;old row=[1, 14];subtract factor=14 times normalized pivot row=[0, 1]

new target row=[1, 0];entry at pivot column1=0

original column family count=2;ambient dimension=3;final exact matrix=[[1, 0], [0, 1], [0, 0]];pivot columns=[0, 1]

rank=2;independent=Yes;spans whole ambient=No;basis=No

supplied original vector count=2;exact rank=2;independence=Yes

this family's lower-bound certificate=dimension_at_least_2

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute dimension eliminate row0 pivot1 column1 entry1 difference with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute dimension eliminate row0 pivot1 column1 entry1 multiple with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute dimension eliminate row0 pivot1 column1 entry0 difference with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute dimension eliminate row0 pivot1 column1 entry0 multiple with exact rational arithmetic, not rounded decimals.
• 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 dimension pivot row1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute dimension pivot row1 column1 normalized entry0 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 dimension pivot row0 column0 normalized entry1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute dimension pivot row0 column0 normalized entry0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(2, 2, Yes, dimension_at_least_2)

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.