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Unformatted text preview: Properties of Matrix Transformations Theorem 4.9.1: For every matrix A the matrix transformation T A : R n R m has the following properties for all vectors u and v in R n and for every scalar k: (a) T A (0) = 0 (b) T A ( ku ) = kT A ( u ) (Homogeneity property) (c) T A ( u + v ) = T A ( u ) + T A ( v ) (Additivity property) (d) T A ( u- v ) = T A ( u )- T A ( v ) Note: We can extend part (c) of Theorem 4.9.1 to three or more vectors. T A ( c 1 v 1 + c 2 v 2 ) = c 1 T A ( v 1 ) + c 2 T A ( v 2 ) (Exercise: show this is true) In fact if v 1 ,v 2 ,...,v k are vectors in R n and c 1 ,c 2 ,...,c k are any scalars then T A ( c 1 v 1 + ... + c k v k ) = c 1 T A ( v 1 ) + ... + c k T A ( v k ) Theorem 4.10.2 T: R n R m is a matrix transformation if and only if the following relationships hold for all vectors u and v in R n and for each scalar k: (a) T(u+v)=T(u)+T(v) (Additivity property) (b) T(ku)=kT(u) (Homogeneity property) Theorem 4.9.2: If T A : R n R m and T B : R n R m are matrix transformations,...
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- Spring '11