# Functional Pairings, Dual Bases and Opposite Coordinate Laws

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

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

batch-20261014-advance-20261105-r02-dual-vector-space-foundations · 2026-10-14 · 100 questions

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

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[1, -1] and primal vector=[2, 3] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([1, -1], [2, 3], -1, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([1, -1], [2, 3], -1, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0001/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 2. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[2, -1] and primal vector=[2, 4] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([2, -1], [2, 4], 0, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([2, -1], [2, 4], 0, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0002/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 3. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[3, -1] and primal vector=[2, 5] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([3, -1], [2, 5], 1, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([3, -1], [2, 5], 1, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0003/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 4. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[4, -1] and primal vector=[2, 6] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([4, -1], [2, 6], 2, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([4, -1], [2, 6], 2, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0004/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 5. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[5, -1] and primal vector=[2, 7] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([5, -1], [2, 7], 3, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([5, -1], [2, 7], 3, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0005/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 6. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[6, -1] and primal vector=[2, 8] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([6, -1], [2, 8], 4, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([6, -1], [2, 8], 4, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0006/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 7. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[7, -1] and primal vector=[2, 9] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([7, -1], [2, 9], 5, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([7, -1], [2, 9], 5, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0007/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 8. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[8, -1] and primal vector=[2, 10] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([8, -1], [2, 10], 6, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([8, -1], [2, 10], 6, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0008/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 9. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[9, -1] and primal vector=[2, 11] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([9, -1], [2, 11], 7, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([9, -1], [2, 11], 7, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0009/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 10. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[10, -1] and primal vector=[2, 12] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([10, -1], [2, 12], 8, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([10, -1], [2, 12], 8, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0010/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 11. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[11, -1] and primal vector=[2, 13] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([11, -1], [2, 13], 9, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([11, -1], [2, 13], 9, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0011/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 12. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[12, -1] and primal vector=[2, 14] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([12, -1], [2, 14], 10, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([12, -1], [2, 14], 10, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0012/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 13. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[13, -1] and primal vector=[2, 15] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([13, -1], [2, 15], 11, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([13, -1], [2, 15], 11, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0013/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 14. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[14, -1] and primal vector=[2, 16] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([14, -1], [2, 16], 12, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([14, -1], [2, 16], 12, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0014/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 15. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[15, -1] and primal vector=[2, 17] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([15, -1], [2, 17], 13, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([15, -1], [2, 17], 13, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0015/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 16. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[16, -1] and primal vector=[2, 18] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([16, -1], [2, 18], 14, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([16, -1], [2, 18], 14, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0016/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 17. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[17, -1] and primal vector=[2, 19] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([17, -1], [2, 19], 15, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([17, -1], [2, 19], 15, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0017/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 18. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[18, -1] and primal vector=[2, 20] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([18, -1], [2, 20], 16, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([18, -1], [2, 20], 16, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0018/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 19. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[19, -1] and primal vector=[2, 21] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([19, -1], [2, 21], 17, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([19, -1], [2, 21], 17, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0019/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 20. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[20, -1] and primal vector=[2, 22] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([20, -1], [2, 22], 18, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([20, -1], [2, 22], 18, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0020/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 21. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[21, -1] and primal vector=[2, 23] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([21, -1], [2, 23], 19, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([21, -1], [2, 23], 19, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0021/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 22. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[22, -1] and primal vector=[2, 24] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([22, -1], [2, 24], 20, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([22, -1], [2, 24], 20, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0022/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 23. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[23, -1] and primal vector=[2, 25] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([23, -1], [2, 25], 21, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([23, -1], [2, 25], 21, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0023/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 24. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[24, -1] and primal vector=[2, 26] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([24, -1], [2, 26], 22, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([24, -1], [2, 26], 22, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0024/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 25. A Covector Evaluates a Vector but Is Not That Vector

Use row coefficients=[25, -1] and primal vector=[2, 27] in one fixed coordinate basis. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (coefficients, vector, value, boundary).

View answer and explanation([25, -1], [2, 27], 23, linear_functional_not_primal_vector)

Evaluate the exact row-column pairing. The row represents an R-valued linear function, not a basis-independent identification with a primal vector. The exact result, retaining the requested order, is ([25, -1], [2, 27], 23, linear_functional_not_primal_vector).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0025/)

Dual vector space · qf-mod36-dual-functional-evaluation · candidate





## 26. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [1, 1]] and candidate dual rows=[[1, -1], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0026/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 27. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [2, 1]] and candidate dual rows=[[1, -2], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0027/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 28. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [3, 1]] and candidate dual rows=[[1, -3], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0028/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 29. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [4, 1]] and candidate dual rows=[[1, -4], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0029/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 30. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [5, 1]] and candidate dual rows=[[1, -5], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0030/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 31. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [6, 1]] and candidate dual rows=[[1, -6], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0031/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 32. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [7, 1]] and candidate dual rows=[[1, -7], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0032/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 33. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [8, 1]] and candidate dual rows=[[1, -8], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0033/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 34. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [9, 1]] and candidate dual rows=[[1, -9], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0034/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 35. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [10, 1]] and candidate dual rows=[[1, -10], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0035/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 36. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [11, 1]] and candidate dual rows=[[1, -11], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0036/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 37. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [12, 1]] and candidate dual rows=[[1, -12], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0037/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 38. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [13, 1]] and candidate dual rows=[[1, -13], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0038/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 39. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [14, 1]] and candidate dual rows=[[1, -14], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0039/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 40. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [15, 1]] and candidate dual rows=[[1, -15], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0040/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 41. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [16, 1]] and candidate dual rows=[[1, -16], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0041/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 42. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [17, 1]] and candidate dual rows=[[1, -17], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0042/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 43. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [18, 1]] and candidate dual rows=[[1, -18], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0043/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 44. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [19, 1]] and candidate dual rows=[[1, -19], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0044/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 45. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [20, 1]] and candidate dual rows=[[1, -20], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0045/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 46. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [21, 1]] and candidate dual rows=[[1, -21], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0046/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 47. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [22, 1]] and candidate dual rows=[[1, -22], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0047/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 48. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [23, 1]] and candidate dual rows=[[1, -23], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0048/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 49. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [24, 1]] and candidate dual rows=[[1, -24], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0049/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 50. Every Dual-Basis Pairing Must Be a Kronecker Delta

Use ordered basis column vectors=[[1, 0], [25, 1]] and candidate dual rows=[[1, -25], [0, 1]]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (pairing_matrix, is_dual_basis, certificate).

View answer and explanation([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta)

Place basis vectors in matrix columns and multiply candidate dual rows by that matrix. The complete pairing matrix, not a few sampled pairings, must be the identity. A failed candidate does not deny existence of the actual dual basis. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], Yes, phi_i_v_j_equals_Kronecker_delta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0050/)

Dual vector space · qf-mod36-dual-dual-basis-certificate · candidate





## 51. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[1, -2], u=[1, 2], v=[3, -1], alpha=1, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-13, -13, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-13, -13, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0051/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 52. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[2, -2], u=[1, 3], v=[4, -1], alpha=2, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-28, -28, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-28, -28, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0052/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 53. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[3, -2], u=[1, 4], v=[5, -1], alpha=3, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-49, -49, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-49, -49, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0053/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 54. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[4, -2], u=[1, 5], v=[6, -1], alpha=4, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-76, -76, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-76, -76, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0054/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 55. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[5, -2], u=[1, 6], v=[7, -1], alpha=5, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-109, -109, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-109, -109, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0055/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 56. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[6, -2], u=[1, 7], v=[8, -1], alpha=6, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-148, -148, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-148, -148, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0056/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 57. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[7, -2], u=[1, 8], v=[9, -1], alpha=7, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-193, -193, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-193, -193, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0057/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 58. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[8, -2], u=[1, 9], v=[10, -1], alpha=8, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-244, -244, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-244, -244, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0058/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 59. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[9, -2], u=[1, 10], v=[11, -1], alpha=9, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-301, -301, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-301, -301, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0059/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 60. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[10, -2], u=[1, 11], v=[12, -1], alpha=10, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-364, -364, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-364, -364, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0060/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 61. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[11, -2], u=[1, 12], v=[13, -1], alpha=11, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-433, -433, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-433, -433, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0061/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 62. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[12, -2], u=[1, 13], v=[14, -1], alpha=12, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-508, -508, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-508, -508, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0062/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 63. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[13, -2], u=[1, 14], v=[15, -1], alpha=13, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-589, -589, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-589, -589, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0063/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 64. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[14, -2], u=[1, 15], v=[16, -1], alpha=14, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-676, -676, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-676, -676, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0064/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 65. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[15, -2], u=[1, 16], v=[17, -1], alpha=15, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-769, -769, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-769, -769, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0065/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 66. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[16, -2], u=[1, 17], v=[18, -1], alpha=16, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-868, -868, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-868, -868, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0066/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 67. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[17, -2], u=[1, 18], v=[19, -1], alpha=17, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-973, -973, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-973, -973, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0067/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 68. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[18, -2], u=[1, 19], v=[20, -1], alpha=18, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-1084, -1084, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1084, -1084, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0068/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 69. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[19, -2], u=[1, 20], v=[21, -1], alpha=19, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-1201, -1201, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1201, -1201, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0069/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 70. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[20, -2], u=[1, 21], v=[22, -1], alpha=20, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-1324, -1324, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1324, -1324, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0070/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 71. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[21, -2], u=[1, 22], v=[23, -1], alpha=21, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-1453, -1453, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1453, -1453, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0071/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 72. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[22, -2], u=[1, 23], v=[24, -1], alpha=22, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-1588, -1588, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1588, -1588, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0072/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 73. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[23, -2], u=[1, 24], v=[25, -1], alpha=23, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-1729, -1729, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1729, -1729, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0073/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 74. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[24, -2], u=[1, 25], v=[26, -1], alpha=24, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-1876, -1876, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-1876, -1876, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0074/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 75. One Row Formula Supplies a Symbolic Linearity Identity

Use covector row=[25, -2], u=[1, 26], v=[27, -1], alpha=25, beta=-2. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (left, right, equal, certificate).

View answer and explanation(-2029, -2029, Yes, row_covector_gives_universal_linearity_certificate)

Compare phi(alpha*u+beta*v) with alpha*phi(u)+beta*phi(v). The row formula supplies the general linearity identity; equality for these displayed inputs alone would not prove linearity of an unspecified function. The exact result, retaining the requested order, is (-2029, -2029, Yes, row_covector_gives_universal_linearity_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0075/)

Dual vector space · qf-mod36-dual-linearity-certificate · candidate





## 76. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 1], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[2, 3] and old covector row=[2, 1]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-1, 3], [2, 3], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-1, 3], [2, 3], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0076/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 77. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 2], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[3, 3] and old covector row=[2, 2]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-3, 3], [2, 6], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-3, 3], [2, 6], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0077/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 78. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 3], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[4, 3] and old covector row=[2, 3]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-5, 3], [2, 9], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-5, 3], [2, 9], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0078/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 79. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 4], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[5, 3] and old covector row=[2, 4]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-7, 3], [2, 12], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-7, 3], [2, 12], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0079/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 80. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 5], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[6, 3] and old covector row=[2, 5]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-9, 3], [2, 15], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-9, 3], [2, 15], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0080/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 81. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 6], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[7, 3] and old covector row=[2, 6]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-11, 3], [2, 18], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-11, 3], [2, 18], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0081/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 82. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 7], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[8, 3] and old covector row=[2, 7]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-13, 3], [2, 21], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-13, 3], [2, 21], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0082/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 83. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 8], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[9, 3] and old covector row=[2, 8]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-15, 3], [2, 24], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-15, 3], [2, 24], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0083/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 84. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 9], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[10, 3] and old covector row=[2, 9]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-17, 3], [2, 27], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-17, 3], [2, 27], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0084/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 85. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 10], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[11, 3] and old covector row=[2, 10]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-19, 3], [2, 30], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-19, 3], [2, 30], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0085/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 86. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 11], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[12, 3] and old covector row=[2, 11]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-21, 3], [2, 33], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-21, 3], [2, 33], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0086/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 87. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 12], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[13, 3] and old covector row=[2, 12]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-23, 3], [2, 36], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-23, 3], [2, 36], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0087/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 88. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 13], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[14, 3] and old covector row=[2, 13]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-25, 3], [2, 39], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-25, 3], [2, 39], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0088/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 89. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 14], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[15, 3] and old covector row=[2, 14]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-27, 3], [2, 42], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-27, 3], [2, 42], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0089/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 90. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 15], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[16, 3] and old covector row=[2, 15]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-29, 3], [2, 45], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-29, 3], [2, 45], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0090/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 91. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 16], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[17, 3] and old covector row=[2, 16]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-31, 3], [2, 48], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-31, 3], [2, 48], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0091/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 92. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 17], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[18, 3] and old covector row=[2, 17]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-33, 3], [2, 51], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-33, 3], [2, 51], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0092/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 93. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 18], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[19, 3] and old covector row=[2, 18]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-35, 3], [2, 54], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-35, 3], [2, 54], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0093/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 94. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 19], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[20, 3] and old covector row=[2, 19]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-37, 3], [2, 57], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-37, 3], [2, 57], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0094/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 95. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 20], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[21, 3] and old covector row=[2, 20]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-39, 3], [2, 60], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-39, 3], [2, 60], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0095/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 96. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 21], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[22, 3] and old covector row=[2, 21]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-41, 3], [2, 63], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-41, 3], [2, 63], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0096/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 97. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 22], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[23, 3] and old covector row=[2, 22]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-43, 3], [2, 66], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-43, 3], [2, 66], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0097/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 98. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 23], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[24, 3] and old covector row=[2, 23]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-45, 3], [2, 69], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-45, 3], [2, 69], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0098/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 99. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 24], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[25, 3] and old covector row=[2, 24]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-47, 3], [2, 72], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-47, 3], [2, 72], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0099/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate





## 100. Primal and Covector Coordinates Follow Opposite Change-of-Basis Laws

Use invertible change matrix P=[[1, 25], [0, 1]] with old_primal=P*new_primal, old primal coordinates=[26, 3] and old covector row=[2, 25]. Coordinates are exact rationals, written as integers or [numerator, denominator] pairs with nonzero denominator. Give (new_primal_coordinates, new_covector_coordinates, coordinate_law, canonical_identification_claim, boundary).

No withholds a canonical identification without extra structure; it does not deny identifications chosen using a basis or inner product.

View answer and explanation([-49, 3], [2, 75], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure)

Compute new_primal=P^(-1)*old_primal and new_covector=old_covector*P, preserving their pairing. The No withholds a canonical V-to-dual identification without extra structure; it does not deny basis-dependent or inner-product-dependent identifications. The exact result, retaining the requested order, is ([-49, 3], [2, 75], contragredient_coordinate_laws, No, no_canonical_V_to_dual_identification_without_extra_structure).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r02-dual-vector-space-0100/)

Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261105-r02-dual-vector-space-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261105-r02-dual-vector-space-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261105-r02-dual-vector-space-foundations/validation-report.json)