# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Linear maps: proving v↦Mv is linear, and Mv+b is linear exactly when b=0](/en/derivations/derivation-mko-linear-transformation-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-linear-transformation)

For every 2×2 real matrix M, prove that T(v)=Mv is a linear map ℝ²→ℝ² (additivity and homogeneity for all vectors u, v and real c); prove that the affine map F(v)=Mv+b is linear exactly when b=0. For the example M₀=[[2,−1],[3,4]], b=(1,−2)ᵀ, u=(1,1)ᵀ, w=(2,−3)ᵀ: compute M₀u, show that F is not linear with an explicit failing equation, and illustrate additivity of T numerically; finally use g(x,y)=(x²,y) to show that finitely many test points cannot prove that an arbitrary map is linear.

Matrix-represented linear maps on the real plane

derivation-mko-linear-transformation-01





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