# Real Plane Matrix Linear Map Foundations 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-20261006-advance-20261007-r02-linear-transformation-foundations · 2026-10-06 · 100 questions This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt. ## 1. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 0], find T(v). View answer and explanation[0, 0] Take the sum of row-by-column products for each component, giving [0, 0]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0001/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 2. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[1, 0], find T(v). View answer and explanation[1, 2] Take the sum of row-by-column products for each component, giving [1, 2]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0002/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 3. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-2, 1], find T(v). View answer and explanation[1, -2] Take the sum of row-by-column products for each component, giving [1, -2]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0003/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 4. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 3], find T(v). View answer and explanation[12, 12] Take the sum of row-by-column products for each component, giving [12, 12]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0004/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 5. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[4, 0], find T(v). View answer and explanation[-32, -8] Take the sum of row-by-column products for each component, giving [-32, -8]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0005/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 6. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-5, 1], find T(v). View answer and explanation[37, 16] Take the sum of row-by-column products for each component, giving [37, 16]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0006/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 7. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 6], find T(v). View answer and explanation[18, 42] Take the sum of row-by-column products for each component, giving [18, 42]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0007/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 8. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[7, 0], find T(v). View answer and explanation[-35, -14] Take the sum of row-by-column products for each component, giving [-35, -14]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0008/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 9. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-8, 1], find T(v). View answer and explanation[33, 25] Take the sum of row-by-column products for each component, giving [33, 25]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0009/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 10. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 9], find T(v). View answer and explanation[18, 90] Take the sum of row-by-column products for each component, giving [18, 90]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0010/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 11. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[10, 0], find T(v). View answer and explanation[-20, -20] Take the sum of row-by-column products for each component, giving [-20, -20]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0011/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 12. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-11, 1], find T(v). View answer and explanation[15, 34] Take the sum of row-by-column products for each component, giving [15, 34]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0012/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 13. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 12], find T(v). View answer and explanation[12, 156] Take the sum of row-by-column products for each component, giving [12, 156]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0013/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 14. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[13, 0], find T(v). View answer and explanation[13, -26] Take the sum of row-by-column products for each component, giving [13, -26]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0014/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 15. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-14, 1], find T(v). View answer and explanation[-25, 43] Take the sum of row-by-column products for each component, giving [-25, 43]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0015/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 16. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 15], find T(v). View answer and explanation[60, 240] Take the sum of row-by-column products for each component, giving [60, 240]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0016/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 17. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[16, 0], find T(v). View answer and explanation[64, -32] Take the sum of row-by-column products for each component, giving [64, -32]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0017/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 18. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-17, 1], find T(v). View answer and explanation[-83, 52] Take the sum of row-by-column products for each component, giving [-83, 52]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0018/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 19. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 18], find T(v). View answer and explanation[54, 342] Take the sum of row-by-column products for each component, giving [54, 342]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0019/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 20. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[19, 0], find T(v). View answer and explanation[133, -38] Take the sum of row-by-column products for each component, giving [133, -38]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0020/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 21. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-20, 1], find T(v). View answer and explanation[-159, 61] Take the sum of row-by-column products for each component, giving [-159, 61]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0021/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 22. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 21], find T(v). View answer and explanation[42, 462] Take the sum of row-by-column products for each component, giving [42, 462]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0022/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 23. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[22, 0], find T(v). View answer and explanation[220, -44] Take the sum of row-by-column products for each component, giving [220, -44]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0023/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 24. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[-23, 1], find T(v). View answer and explanation[-249, 70] Take the sum of row-by-column products for each component, giving [-249, 70]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0024/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 25. Vector Image under a 2×2 Matrix Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For v=[0, 24], find T(v). View answer and explanation[24, 600] Take the sum of row-by-column products for each component, giving [24, 600]. A zero input has zero image; the output is a vector, not a matrix or a guessed point. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0025/) Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate ## 26. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[0, 0]. Is F linear on the whole R²? View answer and explanationYes Here F(0)=b=[0, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving Yes. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0026/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 27. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-1, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-1, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0027/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 28. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-2, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-2, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0028/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 29. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-3, 5]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-3, 5]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0029/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 30. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-4, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-4, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0030/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 31. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-5, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-5, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0031/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 32. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-6, 8]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-6, 8]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0032/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 33. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-7, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-7, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0033/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 34. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-8, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-8, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0034/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 35. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-9, 11]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-9, 11]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0035/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 36. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-10, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-10, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0036/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 37. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-11, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-11, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0037/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 38. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-12, 14]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-12, 14]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0038/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 39. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-13, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-13, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0039/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 40. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-14, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-14, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0040/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 41. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-15, 17]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-15, 17]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0041/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 42. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-16, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-16, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0042/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 43. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-17, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-17, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0043/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 44. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-18, 20]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-18, 20]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0044/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 45. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-19, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-19, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0045/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 46. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-20, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-20, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0046/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 47. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-21, 23]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-21, 23]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0047/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 48. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-22, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-22, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0048/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 49. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-23, 0]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-23, 0]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0049/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 50. Linearity of an Explicit Affine Map Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. The affine map F(x)=Mx+b has b=[-24, 26]. Is F linear on the whole R²? View answer and explanationNo Here F(0)=b=[-24, 26]. For this explicitly specified affine form, linearity holds exactly when b=0, giving No. No invertibility condition on M is required; this is not merely a test at a pair of nonzero inputs. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0050/) Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate ## 51. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[0, 1] and v=[0, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[0, 0] = [0, 0] Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0051/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 52. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[1, 1] and v=[-1, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[4, 8] = [4, 8] Evaluate the two expressions exactly, obtaining [4, 8] = [4, 8]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0052/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 53. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[2, 1] and v=[-2, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[1, 0] = [1, 0] Evaluate the two expressions exactly, obtaining [1, 0] = [1, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0053/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 54. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[3, 1] and v=[-3, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[0, 0] = [0, 0] Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0054/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 55. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[4, 1] and v=[-4, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[3, 15] = [3, 15] Evaluate the two expressions exactly, obtaining [3, 15] = [3, 15]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0055/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 56. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[5, 1] and v=[-5, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[4, 12] = [4, 12] Evaluate the two expressions exactly, obtaining [4, 12] = [4, 12]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0056/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 57. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[6, 1] and v=[-6, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[3, 7] = [3, 7] Evaluate the two expressions exactly, obtaining [3, 7] = [3, 7]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0057/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 58. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[7, 1] and v=[-7, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[0, 0] = [0, 0] Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0058/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 59. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[8, 1] and v=[-8, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[3, 27] = [3, 27] Evaluate the two expressions exactly, obtaining [3, 27] = [3, 27]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0059/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 60. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[9, 1] and v=[-9, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[4, 20] = [4, 20] Evaluate the two expressions exactly, obtaining [4, 20] = [4, 20]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0060/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 61. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[10, 1] and v=[-10, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[3, 11] = [3, 11] Evaluate the two expressions exactly, obtaining [3, 11] = [3, 11]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0061/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 62. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[11, 1] and v=[-11, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[0, 0] = [0, 0] Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0062/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 63. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[12, 1] and v=[-12, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[3, 39] = [3, 39] Evaluate the two expressions exactly, obtaining [3, 39] = [3, 39]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0063/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 64. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[13, 1] and v=[-13, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[4, 28] = [4, 28] Evaluate the two expressions exactly, obtaining [4, 28] = [4, 28]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0064/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 65. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[14, 1] and v=[-14, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[3, 15] = [3, 15] Evaluate the two expressions exactly, obtaining [3, 15] = [3, 15]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0065/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 66. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[15, 1] and v=[-15, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[0, 0] = [0, 0] Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0066/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 67. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[16, 1] and v=[-16, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[3, 51] = [3, 51] Evaluate the two expressions exactly, obtaining [3, 51] = [3, 51]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0067/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 68. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[17, 1] and v=[-17, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[4, 36] = [4, 36] Evaluate the two expressions exactly, obtaining [4, 36] = [4, 36]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0068/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 69. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[18, 1] and v=[-18, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[3, 19] = [3, 19] Evaluate the two expressions exactly, obtaining [3, 19] = [3, 19]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0069/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 70. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[19, 1] and v=[-19, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[0, 0] = [0, 0] Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0070/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 71. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[20, 1] and v=[-20, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[3, 63] = [3, 63] Evaluate the two expressions exactly, obtaining [3, 63] = [3, 63]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0071/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 72. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[21, 1] and v=[-21, 1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[4, 44] = [4, 44] Evaluate the two expressions exactly, obtaining [4, 44] = [4, 44]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0072/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 73. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[22, 1] and v=[-22, 0], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[3, 23] = [3, 23] Evaluate the two expressions exactly, obtaining [3, 23] = [3, 23]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0073/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 74. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[23, 1] and v=[-23, -1], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[0, 0] = [0, 0] Evaluate the two expressions exactly, obtaining [0, 0] = [0, 0]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0074/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 75. Both Values in a Matrix Additivity Identity Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[24, 1] and v=[-24, 2], compute T(u+v) and T(u)+T(v). Answer as left vector = right vector. View answer and explanation[3, 75] = [3, 75] Evaluate the two expressions exactly, obtaining [3, 75] = [3, 75]. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0075/) Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate ## 76. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-12, 1] and c=0, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[0, 0] = [0, 0] Evaluate both sides in the declared model to obtain [0, 0] = [0, 0], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0076/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 77. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[1, 2], [2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[0, 0] and c=-2, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[0, 0] = [0, 0] Evaluate both sides in the declared model to obtain [0, 0] = [0, 0], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0077/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 78. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[0, 1], [1, 0]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-10, 3] and c=3, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[9, -30] = [9, -30] Evaluate both sides in the declared model to obtain [9, -30] = [9, -30], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0078/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 79. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[-9, 4], [-2, 4]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-9, 4] and c=-4, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[-388, -136] = [-388, -136] Evaluate both sides in the declared model to obtain [-388, -136] = [-388, -136], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0079/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 80. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[-8, 1], [-2, 5]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-8, 5] and c=5, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[345, 205] = [345, 205] Evaluate both sides in the declared model to obtain [345, 205] = [345, 205], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0080/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 81. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[-7, 2], [-2, 6]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-7, 6] and c=-1, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[-61, -50] = [-61, -50] Evaluate both sides in the declared model to obtain [-61, -50] = [-61, -50], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0081/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 82. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[-6, 3], [-2, 7]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-6, 7] and c=2, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[114, 122] = [114, 122] Evaluate both sides in the declared model to obtain [114, 122] = [114, 122], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0082/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 83. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[-5, 4], [-2, 8]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-5, 8] and c=-3, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[-171, -222] = [-171, -222] Evaluate both sides in the declared model to obtain [-171, -222] = [-171, -222], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0083/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 84. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[-4, 1], [-2, 9]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-4, 9] and c=4, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[100, 356] = [100, 356] Evaluate both sides in the declared model to obtain [100, 356] = [100, 356], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0084/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 85. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[-3, 2], [-2, 10]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-3, 10] and c=-5, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[-145, -530] = [-145, -530] Evaluate both sides in the declared model to obtain [-145, -530] = [-145, -530], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0085/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 86. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[-2, 3], [-2, 11]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-2, 11] and c=1, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[37, 125] = [37, 125] Evaluate both sides in the declared model to obtain [37, 125] = [37, 125], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0086/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 87. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[-1, 4], [-2, 12]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[-1, 12] and c=-2, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[-98, -292] = [-98, -292] Evaluate both sides in the declared model to obtain [-98, -292] = [-98, -292], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0087/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 88. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[0, 1], [-2, 13]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[0, 13] and c=3, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[39, 507] = [39, 507] Evaluate both sides in the declared model to obtain [39, 507] = [39, 507], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0088/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 89. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[1, 2], [-2, 14]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[1, 14] and c=-4, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[-116, -776] = [-116, -776] Evaluate both sides in the declared model to obtain [-116, -776] = [-116, -776], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0089/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 90. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[2, 3], [-2, 15]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[2, 15] and c=5, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[245, 1105] = [245, 1105] Evaluate both sides in the declared model to obtain [245, 1105] = [245, 1105], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0090/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 91. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[3, 4], [-2, 16]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[3, 16] and c=-1, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[-73, -250] = [-73, -250] Evaluate both sides in the declared model to obtain [-73, -250] = [-73, -250], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0091/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 92. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[4, 1], [-2, 17]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[4, 17] and c=2, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[66, 562] = [66, 562] Evaluate both sides in the declared model to obtain [66, 562] = [66, 562], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0092/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 93. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[5, 2], [-2, 18]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[5, 18] and c=-3, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[-183, -942] = [-183, -942] Evaluate both sides in the declared model to obtain [-183, -942] = [-183, -942], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0093/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 94. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[6, 3], [-2, 19]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[6, 19] and c=4, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[372, 1396] = [372, 1396] Evaluate both sides in the declared model to obtain [372, 1396] = [372, 1396], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0094/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 95. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[7, 4], [-2, 20]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[7, 20] and c=-5, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[-645, -1930] = [-645, -1930] Evaluate both sides in the declared model to obtain [-645, -1930] = [-645, -1930], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0095/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 96. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[8, 1], [-2, 21]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[8, 21] and c=1, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[85, 425] = [85, 425] Evaluate both sides in the declared model to obtain [85, 425] = [85, 425], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0096/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 97. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[9, 2], [-2, 22]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[9, 22] and c=-2, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[-250, -932] = [-250, -932] Evaluate both sides in the declared model to obtain [-250, -932] = [-250, -932], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0097/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 98. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[10, 3], [-2, 23]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[10, 23] and c=3, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[507, 1527] = [507, 1527] Evaluate both sides in the declared model to obtain [507, 1527] = [507, 1527], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0098/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 99. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[11, 4], [-2, 24]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[11, 24] and c=-4, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[-868, -2216] = [-868, -2216] Evaluate both sides in the declared model to obtain [-868, -2216] = [-868, -2216], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0099/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate ## 100. Both Values in a Matrix Scalar Identity Over R, the 2×2 matrix is M=[[12, 1], [-2, 25]]. In the fixed standard basis, T:R²→R² is T(x)=Mx, with column vectors. For u=[12, 25] and c=5, compute T(cu) and cT(u). Answer as left vector = right vector. View answer and explanation[845, 3005] = [845, 3005] Evaluate both sides in the declared model to obtain [845, 3005] = [845, 3005], retaining zero and negative scalars and zero inputs. These are values in the explicit matrix model, not a finite-sample proof of linearity for an arbitrary function. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261006-advance-20261007-r02-linear-transformation-0100/) Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate [Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261006-advance-20261007-r02-linear-transformation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261006-advance-20261007-r02-linear-transformation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261006-advance-20261007-r02-linear-transformation-foundations/validation-report.json)