# Question 16976: A Spanning Family May Give a Loose Upper Bound [Return to catalogue page 85](https://ocme.evemisslab.com/en/questions/page/85/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261102-r01-vector-space-dimension-foundations/#q-20261014-advance-20261102-r01-vector-space-dimension-0076) 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. R0032-16976 · q-20261014-advance-20261102-r01-vector-space-dimension-0076 ## Original question and givens Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [1, 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], [1, 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. ## 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](https://ocme.evemisslab.com/en/mko/mko-vector-space-dimension/) ## Step-by-step derivation S001 · impliesBefore same usual real ambient R^2;original ORDERED vectors as columns=[[1, 0], [0, 1], [1, 1]] After complete row-coordinate matrix=[[1, 0, 1], [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 dependenciesRule: dimension_actual_columns_not_semantic_deduplication · Uses: G001 S002 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore dimension pivot row0 column0 normalized entry2 := div(1, 1) After dimension pivot row0 column0 normalized entry2 = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore pivot row=0;pivot column=0;nonzero exact pivot=1;old row=[1, 0, 1] After divide whole row by pivot;normalized row=[1, 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 dependenciesRule: dimension_nonzero_pivot_scaling_every_entry · Uses: G001, S004 S006 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore 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 dependenciesRule: dimension_nonzero_pivot_scaling_every_entry · Uses: G001, S008 S010 · impliesBefore original column family count=3;ambient dimension=2;final exact matrix=[[1, 0, 1], [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 dependenciesRule: dimension_pivots_relations_and_whole_ambient_spanning · Uses: G001, S009 S011 · impliesBefore 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 dependenciesRule: dimension_spanning_family_only_its_possibly_loose_upper_bound · Uses: G001, S010 ## Reverse checks - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Recompute dimension pivot row1 column1 normalized entry2 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute dimension pivot row1 column1 normalized entry1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute dimension pivot row1 column1 normalized entry0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute dimension pivot row0 column0 normalized entry2 with exact rational arithmetic, not rounded decimals. - 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 (3, 2, Yes, dimension_at_most_3) 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: 326927e1817147c574e674fa37055e213d7a41976033df6e902f47a5163779c2 - candidate record SHA-256: 152db8627e8ef184b365ad693f1b18fd9a0a36b9349238da169ec7eb20453312 - semantic review receipt SHA-256: d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e - question-first witness SHA-256: 790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0076.json)