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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Dual vector space · qf-mod36-dual-coordinate-change-boundary · candidate
Canonical question JSONL · English JSONL · Validation report