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Let V, W, and Y be vector spaces. Suppose dim(V) dim(W) = dim(Y) = 2. = Let ("beta") be an ordered basis for V. Let ("gamma") be an ordered basis for W. Let ("zeta") be an ordered basis for Y. Suppose S is a linear transformation from V to W and that T is a linear trans- formation from W to Y. Remember that ToS is the function from V to Y defined by (TOS)(v) = T(S(v)). (a) Prove that To S is a linear transformation. (b) Prove that ° [T • S] = [T]{[S]}.

Let V, W, and Y be vector spaces. Suppose dim(V) dim(W) = dim(Y) = 2. = Let ("beta") be an ordered basis for V. Let ("gamma") be an ordered basis for W. Let ("zeta") be an ordered basis for Y. Suppose S is a linear transformation from V to W and that T is a linear trans- formation from W to Y. Remember that ToS is the function from V to Y defined by (TOS)(v) = T(S(v)). (a) Prove that To S is a linear transformation. (b) Prove that ° [T • S] = [T]{[S]}.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.5: Basis And Dimension
Problem 69E: Find a basis for R2 that includes the vector (2,2).
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Let V, W, and Y be vector spaces. Suppose dim(V) dim(W) = dim(Y) = 2. = Let ("beta") be an ordered basis for V. Let ("gamma") be an ordered basis for W. Let ("zeta") be an ordered basis for Y. Suppose S is a linear transformation from V to W and that T is a linear trans- formation from W to Y. Remember that ToS is the function from V to Y defined by (TOS)(v) = T(S(v)). (a) Prove that To S is a linear transformation. (b) Prove that ° [T • S] = [T]{[S]}.
Transcribed Image Text:Let V, W, and Y be vector spaces. Suppose dim(V) dim(W) = dim(Y) = 2. = Let ("beta") be an ordered basis for V. Let ("gamma") be an ordered basis for W. Let ("zeta") be an ordered basis for Y. Suppose S is a linear transformation from V to W and that T is a linear trans- formation from W to Y. Remember that ToS is the function from V to Y defined by (TOS)(v) = T(S(v)). (a) Prove that To S is a linear transformation. (b) Prove that ° [T • S] = [T]{[S]}.
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