1. Let V=span (cos x, sin x)F (IR, IR) be the subspace of the real functions...

 

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1. Let V=span (cos x, sin x)F (IR, IR) be the subspace of the real functions that contains linear combinations of sin x and cos x. Let T be the transformation that takes a function f=acosx+b sin x in V to the vector 2f(x/4) EIR 2f(/6) a. Is T a linear transformation? b. If T is a linear transformation, find im(T) and rank(T). c. If T is a linear transformation, find ker(T) and null(T). d. Is T an isomorphism? Are V and R isomorphic? e. Can you find a matrix for T using (sin x, cos x) as a basis for V and (BA) as a basis for R? 1. Let V=span (cos x, sin x)F (IR, IR) be the subspace of the real functions that contains linear combinations of sin x and cos x. Let T be the transformation that takes a function f=acosx+b sin x in V to the vector 2f(x/4) EIR 2f(/6) a. Is T a linear transformation? b. If T is a linear transformation, find im(T) and rank(T). c. If T is a linear transformation, find ker(T) and null(T). d. Is T an isomorphism? Are V and R isomorphic? e. Can you find a matrix for T using (sin x, cos x) as a basis for V and (BA) as a basis for R?

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a To show that T is a linear transformation we need to show that it preserves addition and scalar mu... View the full answer