MAT211_Lecture12[1]

MAT211_Lecture12[1] - Denition MAT211 Lecture 12 Linear...

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1 MAT211 Lecture 12 Linear Transformations and isomorphisms •Linear transformations, image, rank, nullity •Isomorphism and isomorphic spaces •Theorem: Coordinate transformations are isomorphisms •Properties of isomorphisms DeFnition Consider two linear spaces V and W. A function T from V to W is called a linear function if for every pair of elements f and g in V, and every scalar k, T( f + g) = T(f)+T(g) T(k.f) = k. T(f) EXAMPLE: ±ind out whether the following transformations from R 2X2 to R 2X2 are linear. T(M)=M 2 T(M)=7M T(M)=P M P -1 where P is 0 1 1 1 EXAMPLE: ±ind out whether the following transformation from R 2X2 to R 3 is linear. T(M)=(a,b,0) where M is a b c d DeFnition The image of a linear transformation T from V to W, denoted by Im T, is the subset of W {T(f) : f in V}. The kernel of a linear transformation T from V to W, denoted by ker T, is the subset of V {f in V : T(f)=0}. DeFniton
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MAT211_Lecture12[1] - Denition MAT211 Lecture 12 Linear...

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