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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-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.
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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).
[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
Matrix-represented linear maps on the real plane · qf-linear-map-image · candidate
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?
Yes
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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?
No
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
Matrix-represented linear maps on the real plane · qf-linear-map-affine-test · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-additivity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[-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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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.
[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
Matrix-represented linear maps on the real plane · qf-linear-map-homogeneity-values · candidate
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