Basis Qualification, Exact Dimension and One-Way Family Bounds
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Dimension of a finite-dimensional vector space · qf-mod33-dimension-spanning-upper-bound · candidate
Canonical question JSONL · English JSONL · Validation report