Let V be the vector space of all functions f : R → R. Consider the...

Let V be the vector space of all functions f : R → R. Consider the subspace W spanned by {sin(x), cos(x), e^x , e^−x}. The function T : W → W given by taking the derivative is a linear transformation

a) B = {sin(x), cos(x), e^x , e^−x} is a basis for W. Find the matrix for T relative to B.

b)Find all the eigenvalues of the matrix you found in the previous part and describe their eigenvectors. (One of the factors of the characteristic polynomial will be λ 2+1. Just ignore this since it has imaginary roots)

d) Use your answer to the previous part to find all the eigenvalues of T and describe their eigenvectors. Check that the functions you found are indeed eigenvectors of T.

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Expert Solution

Colby Messinger answered 6 months ago

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