Exact Independence, Spanning and Ordered Plane-Basis Coordinates
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batch-20261013-advance-20261024-r01-linear-independence-basis-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-12, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-12, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-12, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
2. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-11, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-11, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-11, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
3. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-10, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-10, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-10, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
4. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-9, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-9, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-9, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
5. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-8, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-8, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-8, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
6. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-7, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-7, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-7, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
7. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-6, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-6, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-6, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
8. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-5, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-5, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-5, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
9. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-4, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-4, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-4, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
10. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-3, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-3, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-3, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
11. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-2, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-2, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-2, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
12. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [-1, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [-1, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [-1, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
13. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [0, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
14. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [1, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [1, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [1, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
15. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [2, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [2, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [2, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
16. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [3, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [3, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [3, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
17. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [4, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [4, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [4, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
18. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [5, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [5, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [5, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
19. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [6, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [6, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [6, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
20. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [7, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [7, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [7, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
21. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [8, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [8, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [8, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
22. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [9, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [9, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [9, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
23. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [10, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [10, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [10, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
24. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [11, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [11, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [11, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
25. Exact Rank versus Independence of a Finite Vector Family
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [12, 1]]. Treat the vectors as columns. Give (original vectors,rank,rank equals number of vectors?).
View answer and explanation
([[1, 0], [12, 1]], 2, Yes)
Independent means that only the all-zero coefficient list gives a zero linear combination. Compare exact rank with the family size, not ambient dimension alone. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [12, 1]], 2, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-independence-audit · candidate
26. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -12], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -12], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -12], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
27. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -11], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -11], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -11], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
28. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -10], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -10], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -10], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
29. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -9], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -9], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -9], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
30. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -8], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -8], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -8], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
31. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -7], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -7], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -7], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
32. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -6], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -6], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -6], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
33. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -5], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -5], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -5], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
34. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -4], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -4], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -4], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
35. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -3], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -3], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -3], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
36. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -2], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -2], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
37. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, -1], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, -1], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
38. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 0], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
39. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 1], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 1], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
40. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 2], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 2], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
41. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 3], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 3], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 3], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
42. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 4], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 4], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 4], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
43. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 5], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 5], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 5], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
44. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 6], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 6], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 6], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
45. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 7], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 7], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 7], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
46. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 8], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 8], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 8], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
47. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 9], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 9], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 9], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
48. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 10], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 10], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 10], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
49. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 11], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 11], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 11], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
50. Independence and Spanning Are Separate Plane-Basis Obligations
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 12], [0, 1]] must all have dimension 2. Give (vectors,rank,independent?,spans R²?,is an R² basis?). Spanning requires rank 2; a basis additionally requires exactly two independent vectors.
View answer and explanation
([[1, 12], [0, 1]], 2, Yes, Yes, Yes)
An independent singleton need not span R², and a redundant spanning family is not a basis. Keep all three flags distinct. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 12], [0, 1]], 2, Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-basis-r2-audit · candidate
51. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -2], [0, 1]], target=[-12, -3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, -2], [0, 1]], [-12, -3], 1, -12, -27, [-12, -3])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-12, -3], 1, -12, -27, [-12, -3]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
52. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -1], [0, 1]], target=[-11, -2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, -1], [0, 1]], [-11, -2], 1, -11, -13, [-11, -2])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-11, -2], 1, -11, -13, [-11, -2]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
53. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 0], [0, 1]], target=[-10, -1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 0], [0, 1]], [-10, -1], 1, -10, -1, [-10, -1])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-10, -1], 1, -10, -1, [-10, -1]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
54. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 1], [0, 1]], target=[-9, 0]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 1], [0, 1]], [-9, 0], 1, -9, 9, [-9, 0])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [-9, 0], 1, -9, 9, [-9, 0]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
55. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[-8, 1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 2], [0, 1]], [-8, 1], 1, -8, 17, [-8, 1])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [-8, 1], 1, -8, 17, [-8, 1]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
56. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -2], [0, 1]], target=[-7, 2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, -2], [0, 1]], [-7, 2], 1, -7, -12, [-7, 2])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-7, 2], 1, -7, -12, [-7, 2]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
57. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -1], [0, 1]], target=[-6, 3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, -1], [0, 1]], [-6, 3], 1, -6, -3, [-6, 3])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-6, 3], 1, -6, -3, [-6, 3]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
58. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 0], [0, 1]], target=[-5, -3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 0], [0, 1]], [-5, -3], 1, -5, -3, [-5, -3])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [-5, -3], 1, -5, -3, [-5, -3]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
59. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 1], [0, 1]], target=[-4, -2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 1], [0, 1]], [-4, -2], 1, -4, 2, [-4, -2])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [-4, -2], 1, -4, 2, [-4, -2]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
60. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[-3, -1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 2], [0, 1]], [-3, -1], 1, -3, 5, [-3, -1])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [-3, -1], 1, -3, 5, [-3, -1]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
61. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -2], [0, 1]], target=[-2, 0]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, -2], [0, 1]], [-2, 0], 1, -2, -4, [-2, 0])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [-2, 0], 1, -2, -4, [-2, 0]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
62. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -1], [0, 1]], target=[-1, 1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, -1], [0, 1]], [-1, 1], 1, -1, 0, [-1, 1])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [-1, 1], 1, -1, 0, [-1, 1]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
63. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 0], [0, 1]], target=[0, 2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 0], [0, 1]], [0, 2], 1, 0, 2, [0, 2])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [0, 2], 1, 0, 2, [0, 2]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
64. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 1], [0, 1]], target=[1, 3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 1], [0, 1]], [1, 3], 1, 1, 2, [1, 3])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [1, 3], 1, 1, 2, [1, 3]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
65. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[2, -3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 2], [0, 1]], [2, -3], 1, 2, -7, [2, -3])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [2, -3], 1, 2, -7, [2, -3]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
66. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -2], [0, 1]], target=[3, -2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, -2], [0, 1]], [3, -2], 1, 3, 4, [3, -2])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [3, -2], 1, 3, 4, [3, -2]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
67. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -1], [0, 1]], target=[4, -1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, -1], [0, 1]], [4, -1], 1, 4, 3, [4, -1])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [4, -1], 1, 4, 3, [4, -1]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
68. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 0], [0, 1]], target=[5, 0]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 0], [0, 1]], [5, 0], 1, 5, 0, [5, 0])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [5, 0], 1, 5, 0, [5, 0]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
69. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 1], [0, 1]], target=[6, 1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 1], [0, 1]], [6, 1], 1, 6, -5, [6, 1])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [6, 1], 1, 6, -5, [6, 1]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
70. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[7, 2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 2], [0, 1]], [7, 2], 1, 7, -12, [7, 2])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [7, 2], 1, 7, -12, [7, 2]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
71. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -2], [0, 1]], target=[8, 3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, -2], [0, 1]], [8, 3], 1, 8, 19, [8, 3])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -2], [0, 1]], [8, 3], 1, 8, 19, [8, 3]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
72. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, -1], [0, 1]], target=[9, -3]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, -1], [0, 1]], [9, -3], 1, 9, 6, [9, -3])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, -1], [0, 1]], [9, -3], 1, 9, 6, [9, -3]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
73. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 0], [0, 1]], target=[10, -2]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 0], [0, 1]], [10, -2], 1, 10, -2, [10, -2])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], [10, -2], 1, 10, -2, [10, -2]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
74. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 1], [0, 1]], target=[11, -1]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 1], [0, 1]], [11, -1], 1, 11, -12, [11, -1])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 1], [0, 1]], [11, -1], 1, 11, -12, [11, -1]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
75. Unique Coordinates Only after Qualifying an Ordered Plane Basis
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. ordered basis=[[1, 2], [0, 1]], target=[12, 0]. Require exactly two independent R² vectors, used as columns. Give (basis,target,signed determinant,first coefficient,second coefficient,reconstructed target). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.
View answer and explanation
([[1, 2], [0, 1]], [12, 0], 1, 12, -24, [12, 0])
Use the nonzero determinant to solve the ordered coordinate system and reconstruct both coordinates. Do not apply square-basis uniqueness to a non-square or dependent family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 2], [0, 1]], [12, 0], 1, 12, -24, [12, 0]).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-coordinate-witness · candidate
76. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -12]], coefficients=[-1, 12, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -12]], [-1, 12, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -12]], [-1, 12, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
77. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -11]], coefficients=[-1, 11, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -11]], [-1, 11, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -11]], [-1, 11, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
78. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -10]], coefficients=[-1, 10, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -10]], [-1, 10, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -10]], [-1, 10, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
79. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -9]], coefficients=[-1, 9, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -9]], [-1, 9, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -9]], [-1, 9, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
80. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -8]], coefficients=[-1, 8, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -8]], [-1, 8, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -8]], [-1, 8, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
81. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -7]], coefficients=[-1, 7, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -7]], [-1, 7, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -7]], [-1, 7, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
82. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -6]], coefficients=[-1, 6, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -6]], [-1, 6, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -6]], [-1, 6, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
83. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -5]], coefficients=[-1, 5, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -5]], [-1, 5, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -5]], [-1, 5, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
84. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -4]], coefficients=[-1, 4, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -4]], [-1, 4, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -4]], [-1, 4, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
85. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -3]], coefficients=[-1, 3, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -3]], [-1, 3, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -3]], [-1, 3, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
86. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -2]], coefficients=[-1, 2, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -2]], [-1, 2, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -2]], [-1, 2, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
87. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, -1]], coefficients=[-1, 1, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, -1]], [-1, 1, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, -1]], [-1, 1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
88. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 0]], coefficients=[-1, 0, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 0]], [-1, 0, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 0]], [-1, 0, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
89. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-1, -1, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 1]], [-1, -1, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-1, -1, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
90. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 2]], coefficients=[-1, -2, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 2]], [-1, -2, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 2]], [-1, -2, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
91. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 3]], coefficients=[-1, -3, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 3]], [-1, -3, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 3]], [-1, -3, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
92. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 4]], coefficients=[-1, -4, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 4]], [-1, -4, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 4]], [-1, -4, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
93. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 5]], coefficients=[-1, -5, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 5]], [-1, -5, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 5]], [-1, -5, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
94. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 6]], coefficients=[-1, -6, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 6]], [-1, -6, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 6]], [-1, -6, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
95. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 7]], coefficients=[-1, -7, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 7]], [-1, -7, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 7]], [-1, -7, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
96. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 8]], coefficients=[-1, -8, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 8]], [-1, -8, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 8]], [-1, -8, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
97. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 9]], coefficients=[-1, -9, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 9]], [-1, -9, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 9]], [-1, -9, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
98. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 10]], coefficients=[-1, -10, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 10]], [-1, -10, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 10]], [-1, -10, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
99. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 11]], coefficients=[-1, -11, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 11]], [-1, -11, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 11]], [-1, -11, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
100. A Zero Combination Must Also Be Nontrivial to Witness Dependence
Use the explicitly supplied finite integer-coordinate vector family in a real coordinate space. Preserve vector order and coefficient pairing. Rank is computed by exact rational elimination; a basis requires both independence and spanning the declared space, not just one condition. vectors=[[1, 0], [0, 1], [1, 12]], coefficients=[-1, -12, 1]. Give (vectors,coefficients,weighted sum,some coefficient nonzero?,sum is zero?,both witness conditions?).
View answer and explanation
([[1, 0], [0, 1], [1, 12]], [-1, -12, 1], [0, 0], Yes, Yes, Yes)
Compute the aligned sum exactly. The trivial all-zero coefficients always give zero but do not witness dependence. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 12]], [-1, -12, 1], [0, 0], Yes, Yes, Yes).
Reviewed step-by-step derivation
Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate
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