# Rectangular Linear Maps, Affine Bias, Additivity and Linear Combinations

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

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

batch-20261014-advance-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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r05-general-linear-map-0100/)

Linear map between real vector spaces · qf-mod27-linearmap-combination-image · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261027-r05-general-linear-map-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261027-r05-general-linear-map-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261027-r05-general-linear-map-foundations/validation-report.json)