Rectangular Linear Maps, Affine Bias, Additivity and Linear Combinations
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261027-r05-general-linear-map-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-12, -3, -5], [-2, 12, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-4, -2, -3]. Give (matrix,vector,image).
View answer and explanation
([[-12, -3, -5], [-2, 12, -6]], [-4, -2, -3], [69, 2])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-12, -3, -5], [-2, 12, -6]], [-4, -2, -3], [69, 2]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
2. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-11, -2, -2], [-1, 11, -1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-3, -1, -2]. Give (matrix,vector,image).
View answer and explanation
([[-11, -2, -2], [-1, 11, -1]], [-3, -1, -2], [39, -6])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-11, -2, -2], [-1, 11, -1]], [-3, -1, -2], [39, -6]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
3. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-10, -1, 1], [0, 10, 4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-2, 0, -1]. Give (matrix,vector,image).
View answer and explanation
([[-10, -1, 1], [0, 10, 4]], [-2, 0, -1], [19, -4])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-10, -1, 1], [0, 10, 4]], [-2, 0, -1], [19, -4]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
4. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-9, 0, 4], [1, 9, -4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-1, 1, 0]. Give (matrix,vector,image).
View answer and explanation
([[-9, 0, 4], [1, 9, -4]], [-1, 1, 0], [9, 8])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-9, 0, 4], [1, 9, -4]], [-1, 1, 0], [9, 8]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
5. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-8, 1, -4], [2, 8, 1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[0, 2, 1]. Give (matrix,vector,image).
View answer and explanation
([[-8, 1, -4], [2, 8, 1]], [0, 2, 1], [-2, 17])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-8, 1, -4], [2, 8, 1]], [0, 2, 1], [-2, 17]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
6. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-7, 2, -1], [-2, 7, 6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[1, -2, 2]. Give (matrix,vector,image).
View answer and explanation
([[-7, 2, -1], [-2, 7, 6]], [1, -2, 2], [-13, -4])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-7, 2, -1], [-2, 7, 6]], [1, -2, 2], [-13, -4]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
7. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-6, 3, 2], [-1, 6, -2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[2, -1, 3]. Give (matrix,vector,image).
View answer and explanation
([[-6, 3, 2], [-1, 6, -2]], [2, -1, 3], [-9, -14])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-6, 3, 2], [-1, 6, -2]], [2, -1, 3], [-9, -14]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
8. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-5, -3, 5], [0, 5, 3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[3, 0, -3]. Give (matrix,vector,image).
View answer and explanation
([[-5, -3, 5], [0, 5, 3]], [3, 0, -3], [-30, -9])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-5, -3, 5], [0, 5, 3]], [3, 0, -3], [-30, -9]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
9. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-4, -2, -3], [1, 4, -5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[4, 1, -2]. Give (matrix,vector,image).
View answer and explanation
([[-4, -2, -3], [1, 4, -5]], [4, 1, -2], [-12, 18])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-4, -2, -3], [1, 4, -5]], [4, 1, -2], [-12, 18]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
10. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-3, -1, 0], [2, 3, 0]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-4, 2, -1]. Give (matrix,vector,image).
View answer and explanation
([[-3, -1, 0], [2, 3, 0]], [-4, 2, -1], [10, -2])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-3, -1, 0], [2, 3, 0]], [-4, 2, -1], [10, -2]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
11. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-2, 0, 3], [-2, 2, 5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-3, -2, 0]. Give (matrix,vector,image).
View answer and explanation
([[-2, 0, 3], [-2, 2, 5]], [-3, -2, 0], [6, 2])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-2, 0, 3], [-2, 2, 5]], [-3, -2, 0], [6, 2]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
12. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[-1, 1, -5], [-1, 1, -3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-2, -1, 1]. Give (matrix,vector,image).
View answer and explanation
([[-1, 1, -5], [-1, 1, -3]], [-2, -1, 1], [-4, -2])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-1, 1, -5], [-1, 1, -3]], [-2, -1, 1], [-4, -2]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
13. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[0, 2, -2], [0, 0, 2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-1, 0, 2]. Give (matrix,vector,image).
View answer and explanation
([[0, 2, -2], [0, 0, 2]], [-1, 0, 2], [-4, 4])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[0, 2, -2], [0, 0, 2]], [-1, 0, 2], [-4, 4]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
14. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[1, 3, 1], [1, -1, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[0, 1, 3]. Give (matrix,vector,image).
View answer and explanation
([[1, 3, 1], [1, -1, -6]], [0, 1, 3], [6, -19])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[1, 3, 1], [1, -1, -6]], [0, 1, 3], [6, -19]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
15. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[2, -3, 4], [2, -2, -1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[1, 2, -3]. Give (matrix,vector,image).
View answer and explanation
([[2, -3, 4], [2, -2, -1]], [1, 2, -3], [-16, 1])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[2, -3, 4], [2, -2, -1]], [1, 2, -3], [-16, 1]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
16. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[3, -2, -4], [-2, -3, 4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[2, -2, -2]. Give (matrix,vector,image).
View answer and explanation
([[3, -2, -4], [-2, -3, 4]], [2, -2, -2], [18, -6])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[3, -2, -4], [-2, -3, 4]], [2, -2, -2], [18, -6]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
17. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[4, -1, -1], [-1, -4, -4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[3, -1, -1]. Give (matrix,vector,image).
View answer and explanation
([[4, -1, -1], [-1, -4, -4]], [3, -1, -1], [14, 5])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[4, -1, -1], [-1, -4, -4]], [3, -1, -1], [14, 5]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
18. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[5, 0, 2], [0, -5, 1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[4, 0, 0]. Give (matrix,vector,image).
View answer and explanation
([[5, 0, 2], [0, -5, 1]], [4, 0, 0], [20, 0])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[5, 0, 2], [0, -5, 1]], [4, 0, 0], [20, 0]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
19. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[6, 1, 5], [1, -6, 6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-4, 1, 1]. Give (matrix,vector,image).
View answer and explanation
([[6, 1, 5], [1, -6, 6]], [-4, 1, 1], [-18, -4])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[6, 1, 5], [1, -6, 6]], [-4, 1, 1], [-18, -4]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
20. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[7, 2, -3], [2, -7, -2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-3, 2, 2]. Give (matrix,vector,image).
View answer and explanation
([[7, 2, -3], [2, -7, -2]], [-3, 2, 2], [-23, -24])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[7, 2, -3], [2, -7, -2]], [-3, 2, 2], [-23, -24]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
21. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[8, 3, 0], [-2, -8, 3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-2, -2, 3]. Give (matrix,vector,image).
View answer and explanation
([[8, 3, 0], [-2, -8, 3]], [-2, -2, 3], [-22, 29])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[8, 3, 0], [-2, -8, 3]], [-2, -2, 3], [-22, 29]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
22. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[9, -3, 3], [-1, -9, -5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[-1, -1, -3]. Give (matrix,vector,image).
View answer and explanation
([[9, -3, 3], [-1, -9, -5]], [-1, -1, -3], [-15, 25])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[9, -3, 3], [-1, -9, -5]], [-1, -1, -3], [-15, 25]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
23. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[10, -2, -5], [0, -10, 0]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[0, 0, -2]. Give (matrix,vector,image).
View answer and explanation
([[10, -2, -5], [0, -10, 0]], [0, 0, -2], [10, 0])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[10, -2, -5], [0, -10, 0]], [0, 0, -2], [10, 0]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
24. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[11, -1, -2], [1, -11, 5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[1, 1, -1]. Give (matrix,vector,image).
View answer and explanation
([[11, -1, -2], [1, -11, 5]], [1, 1, -1], [12, -15])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[11, -1, -2], [1, -11, 5]], [1, 1, -1], [12, -15]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
25. Evaluate a Possibly Rectangular Real Linear Map
Use the complete rectangular matrix A=[[12, 0, 1], [2, -12, -3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. vector=[2, 2, 0]. Give (matrix,vector,image).
View answer and explanation
([[12, 0, 1], [2, -12, -3]], [2, 2, 0], [24, -20])
Take each row dot product with the input vector, retaining the row/column direction and original order. The input and output dimensions need not be equal. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[12, 0, 1], [2, -12, -3]], [2, 2, 0], [24, -20]).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-matrix-evaluate · candidate
26. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-12, -3, -5], [-2, 12, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[0, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-12, -3, -5], [-2, 12, -6]], [0, 0], [0, 0], Yes, matrix_linear_certificate)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-12, -3, -5], [-2, 12, -6]], [0, 0], [0, 0], Yes, matrix_linear_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
27. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-11, -2, -2], [-1, 11, -1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[2, -2]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-11, -2, -2], [-1, 11, -1]], [2, -2], [2, -2], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-11, -2, -2], [-1, 11, -1]], [2, -2], [2, -2], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
28. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-10, -1, 1], [0, 10, 4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[3, -1]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-10, -1, 1], [0, 10, 4]], [3, -1], [3, -1], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-10, -1, 1], [0, 10, 4]], [3, -1], [3, -1], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
29. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-9, 0, 4], [1, 9, -4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[4, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-9, 0, 4], [1, 9, -4]], [4, 0], [4, 0], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-9, 0, 4], [1, 9, -4]], [4, 0], [4, 0], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
30. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-8, 1, -4], [2, 8, 1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[0, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-8, 1, -4], [2, 8, 1]], [0, 0], [0, 0], Yes, matrix_linear_certificate)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-8, 1, -4], [2, 8, 1]], [0, 0], [0, 0], Yes, matrix_linear_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
31. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-7, 2, -1], [-2, 7, 6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[1, 2]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-7, 2, -1], [-2, 7, 6]], [1, 2], [1, 2], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-7, 2, -1], [-2, 7, 6]], [1, 2], [1, 2], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
32. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-6, 3, 2], [-1, 6, -2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[2, 3]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-6, 3, 2], [-1, 6, -2]], [2, 3], [2, 3], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-6, 3, 2], [-1, 6, -2]], [2, 3], [2, 3], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
33. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-5, -3, 5], [0, 5, 3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[3, -3]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-5, -3, 5], [0, 5, 3]], [3, -3], [3, -3], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-5, -3, 5], [0, 5, 3]], [3, -3], [3, -3], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
34. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-4, -2, -3], [1, 4, -5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[0, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-4, -2, -3], [1, 4, -5]], [0, 0], [0, 0], Yes, matrix_linear_certificate)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-4, -2, -3], [1, 4, -5]], [0, 0], [0, 0], Yes, matrix_linear_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
35. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-3, -1, 0], [2, 3, 0]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[5, -1]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-3, -1, 0], [2, 3, 0]], [5, -1], [5, -1], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-3, -1, 0], [2, 3, 0]], [5, -1], [5, -1], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
36. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-2, 0, 3], [-2, 2, 5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[1, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-2, 0, 3], [-2, 2, 5]], [1, 0], [1, 0], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-2, 0, 3], [-2, 2, 5]], [1, 0], [1, 0], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
37. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[-1, 1, -5], [-1, 1, -3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[2, 1]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[-1, 1, -5], [-1, 1, -3]], [2, 1], [2, 1], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[-1, 1, -5], [-1, 1, -3]], [2, 1], [2, 1], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
38. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[0, 2, -2], [0, 0, 2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[0, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[0, 2, -2], [0, 0, 2]], [0, 0], [0, 0], Yes, matrix_linear_certificate)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[0, 2, -2], [0, 0, 2]], [0, 0], [0, 0], Yes, matrix_linear_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
39. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[1, 3, 1], [1, -1, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[4, 3]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[1, 3, 1], [1, -1, -6]], [4, 3], [4, 3], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[1, 3, 1], [1, -1, -6]], [4, 3], [4, 3], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
40. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[2, -3, 4], [2, -2, -1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[5, -3]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[2, -3, 4], [2, -2, -1]], [5, -3], [5, -3], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[2, -3, 4], [2, -2, -1]], [5, -3], [5, -3], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
41. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[3, -2, -4], [-2, -3, 4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[1, -2]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[3, -2, -4], [-2, -3, 4]], [1, -2], [1, -2], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[3, -2, -4], [-2, -3, 4]], [1, -2], [1, -2], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
42. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[4, -1, -1], [-1, -4, -4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[0, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[4, -1, -1], [-1, -4, -4]], [0, 0], [0, 0], Yes, matrix_linear_certificate)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[4, -1, -1], [-1, -4, -4]], [0, 0], [0, 0], Yes, matrix_linear_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
43. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[5, 0, 2], [0, -5, 1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[3, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[5, 0, 2], [0, -5, 1]], [3, 0], [3, 0], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[5, 0, 2], [0, -5, 1]], [3, 0], [3, 0], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
44. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[6, 1, 5], [1, -6, 6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[4, 1]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[6, 1, 5], [1, -6, 6]], [4, 1], [4, 1], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[6, 1, 5], [1, -6, 6]], [4, 1], [4, 1], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
45. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[7, 2, -3], [2, -7, -2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[5, 2]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[7, 2, -3], [2, -7, -2]], [5, 2], [5, 2], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[7, 2, -3], [2, -7, -2]], [5, 2], [5, 2], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
46. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[8, 3, 0], [-2, -8, 3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[0, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[8, 3, 0], [-2, -8, 3]], [0, 0], [0, 0], Yes, matrix_linear_certificate)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[8, 3, 0], [-2, -8, 3]], [0, 0], [0, 0], Yes, matrix_linear_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
47. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[9, -3, 3], [-1, -9, -5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[2, -3]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[9, -3, 3], [-1, -9, -5]], [2, -3], [2, -3], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[9, -3, 3], [-1, -9, -5]], [2, -3], [2, -3], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
48. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[10, -2, -5], [0, -10, 0]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[3, -2]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[10, -2, -5], [0, -10, 0]], [3, -2], [3, -2], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[10, -2, -5], [0, -10, 0]], [3, -2], [3, -2], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
49. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[11, -1, -2], [1, -11, 5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[4, -1]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[11, -1, -2], [1, -11, 5]], [4, -1], [4, -1], No, nonzero_translation_is_not_linear)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[11, -1, -2], [1, -11, 5]], [4, -1], [4, -1], No, nonzero_translation_is_not_linear).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
50. A Nonzero Affine Translation Fails the Zero-Image Obligation
Use the complete rectangular matrix A=[[12, 0, 1], [2, -12, -3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. The actual map is T(v)=A*v+bias, with bias=[0, 0]. Give (matrix,bias,T_zero,linear?,boundary). A zero bias yields matrix_linear_certificate; a nonzero bias yields nonzero_translation_is_not_linear.
View answer and explanation
([[12, 0, 1], [2, -12, -3]], [0, 0], [0, 0], Yes, matrix_linear_certificate)
Evaluate T at the zero vector in the input space. A nonzero bias immediately refutes linearity; a zero bias leaves the universal matrix formula. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([[12, 0, 1], [2, -12, -3]], [0, 0], [0, 0], Yes, matrix_linear_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-affine-qualification · candidate
51. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-12, -3, -5], [-2, 12, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[0, 0], u=[1, -2, -3], v=[-5, 1, -2]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, -2, -3], [-5, 1, -2], [0, 0], [76, 26], [76, 26], [0, 0], Yes)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, -2, -3], [-5, 1, -2], [0, 0], [76, 26], [76, 26], [0, 0], Yes).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
52. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-11, -2, -2], [-1, 11, -1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[2, -2], u=[1, -1, -2], v=[-4, 1, -1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, -1, -2], [-4, 1, -1], [2, -2], [41, 4], [43, 2], [-2, 2], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, -1, -2], [-4, 1, -1], [2, -2], [41, 4], [43, 2], [-2, 2], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
53. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-10, -1, 1], [0, 10, 4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[3, -1], u=[1, 0, -1], v=[-3, 1, 0]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 0, -1], [-3, 1, 0], [3, -1], [21, 5], [24, 4], [-3, 1], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 0, -1], [-3, 1, 0], [3, -1], [21, 5], [24, 4], [-3, 1], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
54. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-9, 0, 4], [1, 9, -4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[4, 0], u=[1, 1, 0], v=[-2, 1, 1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 1, 0], [-2, 1, 1], [4, 0], [17, 13], [21, 13], [-4, 0], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 1, 0], [-2, 1, 1], [4, 0], [17, 13], [21, 13], [-4, 0], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
55. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-8, 1, -4], [2, 8, 1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[0, 0], u=[1, 2, 1], v=[-1, 1, -2]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 2, 1], [-1, 1, -2], [0, 0], [7, 23], [7, 23], [0, 0], Yes)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, 1], [-1, 1, -2], [0, 0], [7, 23], [7, 23], [0, 0], Yes).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
56. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-7, 2, -1], [-2, 7, 6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[1, 2], u=[1, -2, 2], v=[0, 1, -1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, -2, 2], [0, 1, -1], [1, 2], [-9, -1], [-8, 1], [-1, -2], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, -2, 2], [0, 1, -1], [1, 2], [-9, -1], [-8, 1], [-1, -2], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
57. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-6, 3, 2], [-1, 6, -2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[2, 3], u=[1, -1, 3], v=[1, 1, 0]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, -1, 3], [1, 1, 0], [2, 3], [-4, -5], [-2, -2], [-2, -3], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, -1, 3], [1, 1, 0], [2, 3], [-4, -5], [-2, -2], [-2, -3], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
58. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-5, -3, 5], [0, 5, 3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[3, -3], u=[1, 0, -3], v=[2, 1, 1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 0, -3], [2, 1, 1], [3, -3], [-25, -4], [-22, -7], [-3, 3], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 0, -3], [2, 1, 1], [3, -3], [-25, -4], [-22, -7], [-3, 3], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
59. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-4, -2, -3], [1, 4, -5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[0, 0], u=[1, 1, -2], v=[3, 1, -2]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 1, -2], [3, 1, -2], [0, 0], [-8, 32], [-8, 32], [0, 0], Yes)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 1, -2], [3, 1, -2], [0, 0], [-8, 32], [-8, 32], [0, 0], Yes).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
60. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-3, -1, 0], [2, 3, 0]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[5, -1], u=[1, 2, -1], v=[4, 1, -1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 2, -1], [4, 1, -1], [5, -1], [-13, 18], [-8, 17], [-5, 1], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, -1], [4, 1, -1], [5, -1], [-13, 18], [-8, 17], [-5, 1], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
61. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-2, 0, 3], [-2, 2, 5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[1, 0], u=[1, -2, 0], v=[5, 1, 0]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, -2, 0], [5, 1, 0], [1, 0], [-11, -14], [-10, -14], [-1, 0], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, -2, 0], [5, 1, 0], [1, 0], [-11, -14], [-10, -14], [-1, 0], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
62. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[-1, 1, -5], [-1, 1, -3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[2, 1], u=[1, -1, 1], v=[-5, 1, 1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, -1, 1], [-5, 1, 1], [2, 1], [-4, -1], [-2, 0], [-2, -1], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, -1, 1], [-5, 1, 1], [2, 1], [-4, -1], [-2, 0], [-2, -1], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
63. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[0, 2, -2], [0, 0, 2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[0, 0], u=[1, 0, 2], v=[-4, 1, -2]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 0, 2], [-4, 1, -2], [0, 0], [2, 0], [2, 0], [0, 0], Yes)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 0, 2], [-4, 1, -2], [0, 0], [2, 0], [2, 0], [0, 0], Yes).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
64. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[1, 3, 1], [1, -1, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[4, 3], u=[1, 1, 3], v=[-3, 1, -1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 1, 3], [-3, 1, -1], [4, 3], [10, -13], [14, -10], [-4, -3], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 1, 3], [-3, 1, -1], [4, 3], [10, -13], [14, -10], [-4, -3], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
65. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[2, -3, 4], [2, -2, -1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[5, -3], u=[1, 2, -3], v=[-2, 1, 0]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 2, -3], [-2, 1, 0], [5, -3], [-18, -8], [-13, -11], [-5, 3], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, -3], [-2, 1, 0], [5, -3], [-18, -8], [-13, -11], [-5, 3], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
66. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[3, -2, -4], [-2, -3, 4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[1, -2], u=[1, -2, -2], v=[-1, 1, 1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, -2, -2], [-1, 1, 1], [1, -2], [7, -3], [8, -5], [-1, 2], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, -2, -2], [-1, 1, 1], [1, -2], [7, -3], [8, -5], [-1, 2], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
67. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[4, -1, -1], [-1, -4, -4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[0, 0], u=[1, -1, -1], v=[0, 1, -2]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, -1, -1], [0, 1, -2], [0, 0], [7, 11], [7, 11], [0, 0], Yes)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, -1, -1], [0, 1, -2], [0, 0], [7, 11], [7, 11], [0, 0], Yes).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
68. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[5, 0, 2], [0, -5, 1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[3, 0], u=[1, 0, 0], v=[1, 1, -1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 0, 0], [1, 1, -1], [3, 0], [11, -6], [14, -6], [-3, 0], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 0, 0], [1, 1, -1], [3, 0], [11, -6], [14, -6], [-3, 0], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
69. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[6, 1, 5], [1, -6, 6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[4, 1], u=[1, 1, 1], v=[2, 1, 0]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 1, 1], [2, 1, 0], [4, 1], [29, -2], [33, -1], [-4, -1], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 1, 1], [2, 1, 0], [4, 1], [29, -2], [33, -1], [-4, -1], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
70. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[7, 2, -3], [2, -7, -2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[5, 2], u=[1, 2, 2], v=[3, 1, 1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 2, 2], [3, 1, 1], [5, 2], [30, -17], [35, -15], [-5, -2], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, 2], [3, 1, 1], [5, 2], [30, -17], [35, -15], [-5, -2], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
71. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[8, 3, 0], [-2, -8, 3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[0, 0], u=[1, -2, 3], v=[4, 1, -2]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, -2, 3], [4, 1, -2], [0, 0], [37, 1], [37, 1], [0, 0], Yes)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, -2, 3], [4, 1, -2], [0, 0], [37, 1], [37, 1], [0, 0], Yes).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
72. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[9, -3, 3], [-1, -9, -5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[2, -3], u=[1, -1, -3], v=[5, 1, -1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, -1, -3], [5, 1, -1], [2, -3], [44, 11], [46, 8], [-2, 3], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, -1, -3], [5, 1, -1], [2, -3], [44, 11], [46, 8], [-2, 3], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
73. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[10, -2, -5], [0, -10, 0]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[3, -2], u=[1, 0, -2], v=[-5, 1, 0]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 0, -2], [-5, 1, 0], [3, -2], [-29, -12], [-26, -14], [-3, 2], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 0, -2], [-5, 1, 0], [3, -2], [-29, -12], [-26, -14], [-3, 2], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
74. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[11, -1, -2], [1, -11, 5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[4, -1], u=[1, 1, -1], v=[-4, 1, 1]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 1, -1], [-4, 1, 1], [4, -1], [-31, -26], [-27, -27], [-4, 1], No)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 1, -1], [-4, 1, 1], [4, -1], [-31, -26], [-27, -27], [-4, 1], No).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
75. Compare Both Sides of Additivity for the Supplied Affine Map
Use the complete rectangular matrix A=[[12, 0, 1], [2, -12, -3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. T(v)=A*v+bias, bias=[0, 0], u=[1, 2, 0], v=[-3, 1, -2]. Give (u,v,bias,T(u+v),T(u)+T(v),lhs−rhs,additive for this instance?).
View answer and explanation
([1, 2, 0], [-3, 1, -2], [0, 0], [-26, -34], [-26, -34], [0, 0], Yes)
The left side contains one bias and the right side contains two, so their difference is the negative bias. Retain that signed vector. No finite passing sample proves arbitrary black-box linearity. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, 0], [-3, 1, -2], [0, 0], [-26, -34], [-26, -34], [0, 0], Yes).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-additivity-witness · candidate
76. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-12, -3, -5], [-2, 12, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -2, -3], v=[-5, 1, -2], alpha=-2, beta=-3; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(-2, -3, [13, 1, 12], [-219, -86], [-219, -86], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -3, [13, 1, 12], [-219, -86], [-219, -86], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
77. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-11, -2, -2], [-1, 11, -1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -1, -2], v=[-4, 1, -1], alpha=-1, beta=-2; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(-1, -2, [7, -1, 4], [-83, -22], [-83, -22], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -2, [7, -1, 4], [-83, -22], [-83, -22], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
78. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-10, -1, 1], [0, 10, 4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 0, -1], v=[-3, 1, 0], alpha=0, beta=-1; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(0, -1, [3, -1, 0], [-29, -10], [-29, -10], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -1, [3, -1, 0], [-29, -10], [-29, -10], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
79. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-9, 0, 4], [1, 9, -4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 1, 0], v=[-2, 1, 1], alpha=1, beta=0; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(1, 0, [1, 1, 0], [-9, 10], [-9, 10], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 0, [1, 1, 0], [-9, 10], [-9, 10], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
80. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-8, 1, -4], [2, 8, 1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 2, 1], v=[-1, 1, -2], alpha=2, beta=1; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(2, 1, [1, 5, 0], [-3, 42], [-3, 42], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 1, [1, 5, 0], [-3, 42], [-3, 42], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
81. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-7, 2, -1], [-2, 7, 6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -2, 2], v=[0, 1, -1], alpha=-2, beta=2; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(-2, 2, [-2, 6, -6], [32, 10], [32, 10], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 2, [-2, 6, -6], [32, 10], [32, 10], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
82. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-6, 3, 2], [-1, 6, -2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -1, 3], v=[1, 1, 0], alpha=-1, beta=3; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(-1, 3, [2, 4, -3], [-6, 28], [-6, 28], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 3, [2, 4, -3], [-6, 28], [-6, 28], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
83. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-5, -3, 5], [0, 5, 3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 0, -3], v=[2, 1, 1], alpha=0, beta=-3; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(0, -3, [-6, -3, -3], [24, -24], [24, -24], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -3, [-6, -3, -3], [24, -24], [24, -24], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
84. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-4, -2, -3], [1, 4, -5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 1, -2], v=[3, 1, -2], alpha=1, beta=-2; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(1, -2, [-5, -1, 2], [16, -19], [16, -19], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -2, [-5, -1, 2], [16, -19], [16, -19], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
85. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-3, -1, 0], [2, 3, 0]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 2, -1], v=[4, 1, -1], alpha=2, beta=-1; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(2, -1, [-2, 3, -1], [3, 5], [3, 5], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -1, [-2, 3, -1], [3, 5], [3, 5], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
86. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-2, 0, 3], [-2, 2, 5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -2, 0], v=[5, 1, 0], alpha=-2, beta=0; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(-2, 0, [-2, 4, 0], [4, 12], [4, 12], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 0, [-2, 4, 0], [4, 12], [4, 12], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
87. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[-1, 1, -5], [-1, 1, -3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -1, 1], v=[-5, 1, 1], alpha=-1, beta=1; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(-1, 1, [-6, 2, 0], [8, 8], [8, 8], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, 1, [-6, 2, 0], [8, 8], [8, 8], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
88. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[0, 2, -2], [0, 0, 2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 0, 2], v=[-4, 1, -2], alpha=0, beta=2; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(0, 2, [-8, 2, -4], [12, -8], [12, -8], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 2, [-8, 2, -4], [12, -8], [12, -8], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
89. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[1, 3, 1], [1, -1, -6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 1, 3], v=[-3, 1, -1], alpha=1, beta=3; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(1, 3, [-8, 4, 0], [4, -12], [4, -12], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 3, [-8, 4, 0], [4, -12], [4, -12], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
90. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[2, -3, 4], [2, -2, -1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 2, -3], v=[-2, 1, 0], alpha=2, beta=-3; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(2, -3, [8, 1, -6], [-11, 20], [-11, 20], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, -3, [8, 1, -6], [-11, 20], [-11, 20], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
91. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[3, -2, -4], [-2, -3, 4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -2, -2], v=[-1, 1, 1], alpha=-2, beta=-2; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(-2, -2, [0, 2, 2], [-12, 2], [-12, 2], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, -2, [0, 2, 2], [-12, 2], [-12, 2], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
92. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[4, -1, -1], [-1, -4, -4]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -1, -1], v=[0, 1, -2], alpha=-1, beta=-1; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(-1, -1, [-1, 0, 3], [-7, -11], [-7, -11], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -1, [-1, 0, 3], [-7, -11], [-7, -11], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
93. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[5, 0, 2], [0, -5, 1]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 0, 0], v=[1, 1, -1], alpha=0, beta=0; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(0, 0, [0, 0, 0], [0, 0], [0, 0], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 0, [0, 0, 0], [0, 0], [0, 0], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
94. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[6, 1, 5], [1, -6, 6]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 1, 1], v=[2, 1, 0], alpha=1, beta=1; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(1, 1, [3, 2, 1], [25, -3], [25, -3], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, 1, [3, 2, 1], [25, -3], [25, -3], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
95. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[7, 2, -3], [2, -7, -2]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 2, 2], v=[3, 1, 1], alpha=2, beta=2; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(2, 2, [8, 6, 6], [50, -38], [50, -38], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 2, [8, 6, 6], [50, -38], [50, -38], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
96. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[8, 3, 0], [-2, -8, 3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -2, 3], v=[4, 1, -2], alpha=-2, beta=3; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(-2, 3, [10, 7, -12], [101, -112], [101, -112], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-2, 3, [10, 7, -12], [101, -112], [101, -112], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
97. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[9, -3, 3], [-1, -9, -5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, -1, -3], v=[5, 1, -1], alpha=-1, beta=-3; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(-1, -3, [-16, -2, 6], [-120, 4], [-120, 4], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (-1, -3, [-16, -2, 6], [-120, 4], [-120, 4], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
98. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[10, -2, -5], [0, -10, 0]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 0, -2], v=[-5, 1, 0], alpha=0, beta=-2; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(0, -2, [10, -2, 0], [104, 20], [104, 20], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, -2, [10, -2, 0], [104, 20], [104, 20], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
99. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[11, -1, -2], [1, -11, 5]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 1, -1], v=[-4, 1, 1], alpha=1, beta=-1; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(1, -1, [5, 0, -2], [59, -5], [59, -5], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1, -1, [5, 0, -2], [59, -5], [59, -5], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
100. The Matrix Formula Preserves a Whole Linear Combination
Use the complete rectangular matrix A=[[12, 0, 1], [2, -12, -3]] as a map from its column-coordinate space to its row-coordinate space, both over the reals. Matrix distributivity is a universal coordinate certificate. All vectors and any bias must have their respective input/output dimensions. Finite samples do not establish linearity of an arbitrary black box, or imply injectivity, surjectivity, boundedness or continuity beyond the stated claim. u=[1, 2, 0], v=[-3, 1, -2], alpha=2, beta=0; no bias is supplied. Give (alpha,beta,combined,lhs=A*combined,rhs=alpha*A*u+beta*A*v,equal?,matrix_formula_universal_certificate).
View answer and explanation
(2, 0, [2, 4, 0], [24, -44], [24, -44], Yes, matrix_formula_universal_certificate)
Use distributivity of each row dot product, then compute the two exact images in the target-coordinate space. The universal certificate comes from the formula, not just this instance. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2, 0, [2, 4, 0], [24, -44], [24, -44], Yes, matrix_formula_universal_certificate).
Reviewed step-by-step derivation
Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate
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