# Basis Qualification, Exact Dimension and One-Way Family Bounds

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261102-r01-vector-space-dimension-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0001/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 2. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0002/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 3. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [2, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0003/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 4. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [3, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0004/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 5. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [4, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0005/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 6. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [5, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0006/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 7. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [6, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0007/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 8. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0008/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 9. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [8, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0009/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 10. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [9, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0010/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 11. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [10, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0011/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 12. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [11, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0012/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 13. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [12, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0013/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 14. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [13, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0014/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 15. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [14, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0015/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 16. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [15, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0016/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 17. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [16, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0017/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 18. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [17, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0018/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 19. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [18, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0019/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 20. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [19, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0020/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 21. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [20, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0021/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 22. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [21, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0022/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 23. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [22, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0023/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 24. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [23, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0024/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 25. A Basis Requires Both Independence and Spanning

Use the complete ordered real coordinate family vectors=[[1, 0], [24, 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 (ambient_dimension,vector_count,rank,independent?,spanning?,basis?).

View answer and explanation(2, 2, 2, Yes, Yes, Yes)

Compare rank with the number of vectors for independence and with the ambient coordinate dimension for spanning. Conjoin both to qualify the supplied family as a basis; neither property alone makes its size the dimension. The exact result, retaining the requested order, is (2, 2, 2, Yes, Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0025/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-basis-audit · candidate





## 26. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0026/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 27. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0027/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 28. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [2, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0028/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 29. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [3, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0029/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 30. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [4, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0030/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 31. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [5, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0031/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 32. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [6, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0032/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 33. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0033/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 34. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [8, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0034/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 35. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [9, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0035/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 36. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [10, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0036/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 37. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [11, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0037/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 38. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [12, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0038/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 39. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [13, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0039/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 40. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [14, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0040/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 41. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [15, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0041/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 42. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [16, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0042/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 43. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [17, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0043/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 44. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [18, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0044/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 45. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [19, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0045/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 46. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [20, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0046/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 47. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [21, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0047/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 48. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [22, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0048/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 49. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [23, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0049/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 50. Read Dimension from a Previously Qualified Basis

Use the complete ordered real coordinate family vectors=[[1, 0], [24, 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. Require this complete family to be an actual basis. Give (ambient_dimension,basis_count,rank,dimension,finite_dimensional_basis_cardinality_invariance_theorem).

View answer and explanation(2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem)

Check independence and spanning before reading the basis cardinality. A dependent or nonspanning family is rejected rather than assigned a dimension from its length. The named invariance theorem is not newly proved by this instance or recorded as Lean Evidence. The exact result, retaining the requested order, is (2, 2, 2, 2, finite_dimensional_basis_cardinality_invariance_theorem).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0050/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-exact-value · candidate





## 51. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [0, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0051/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 52. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [1, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0052/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 53. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [2, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0053/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 54. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [3, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0054/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 55. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [4, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0055/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 56. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [5, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0056/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 57. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [6, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0057/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 58. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [7, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0058/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 59. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [8, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0059/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 60. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [9, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0060/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 61. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [10, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0061/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 62. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [11, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0062/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 63. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [12, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0063/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 64. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [13, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0064/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 65. An Independent Family Gives Only a Lower Bound

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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0065/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 66. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [15, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0066/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 67. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [16, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0067/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 68. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [17, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0068/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 69. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [18, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0069/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 70. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [19, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0070/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 71. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [20, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0071/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 72. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [21, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0072/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 73. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [22, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0073/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 74. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [23, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0074/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 75. An Independent Family Gives Only a Lower Bound

Use the complete ordered real coordinate family vectors=[[1, 0, 0], [24, 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.

View answer and explanation(2, 2, Yes, dimension_at_least_2)

Only a verified independent family supplies this size-based lower bound. If this family is dependent, the result withholds its certificate; it does not refute every possible lower bound for the ambient space. The exact result, retaining the requested order, is (2, 2, Yes, dimension_at_least_2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0075/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-independent-lower-bound · candidate





## 76. A Spanning Family May Give a Loose Upper Bound

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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0076/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 77. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [2, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0077/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 78. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [3, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0078/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 79. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [4, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0079/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 80. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [5, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0080/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 81. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [6, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0081/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 82. A Spanning Family May Give a Loose Upper Bound

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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0082/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 83. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [8, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0083/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 84. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [9, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0084/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 85. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [10, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0085/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 86. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [11, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0086/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 87. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [12, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0087/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 88. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [13, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0088/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 89. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [14, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0089/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 90. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [15, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0090/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 91. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [16, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0091/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 92. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [17, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0092/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 93. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [18, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0093/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 94. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [19, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0094/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 95. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [20, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0095/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 96. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [21, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0096/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 97. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [22, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0097/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 98. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [23, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0098/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 99. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [24, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0099/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate





## 100. A Spanning Family May Give a Loose Upper Bound

Use the complete ordered real coordinate family vectors=[[1, 0], [0, 1], [25, 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.

View answer and explanation(3, 2, Yes, dimension_at_most_3)

Only a verified spanning family supplies this upper bound. A redundant spanning family can be larger than the exact dimension; absence of a certificate from this family does not prove that the space is infinite-dimensional. The exact result, retaining the requested order, is (3, 2, Yes, dimension_at_most_3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261102-r01-vector-space-dimension-0100/)

Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261102-r01-vector-space-dimension-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261102-r01-vector-space-dimension-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261102-r01-vector-space-dimension-foundations/validation-report.json)