# Exact Independence, Spanning and Ordered Plane-Basis Coordinates

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

batch-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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r01-linear-independence-basis-0100/)

Linear independence and bases · qf-mod24-linbasis-dependence-witness-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261024-r01-linear-independence-basis-foundations/validation-report.json)