Let V be the vector space of all functions f : R ? R. Considerthe subspace W spanned by {sin(x), cos(x), e^x , e^?x}. Thefunction T : W ? W given by taking the derivative is a lineartransformation
a) B = {sin(x), cos(x), e^x , e^?x} is a basis for W. Find thematrix for T relative to B.
b)Find all the eigenvalues of the matrix you found in theprevious part and describe their eigenvectors. (One of the factorsof the characteristic polynomial will be ? 2+1. Just ignore thissince it has imaginary roots)
d) Use your answer to the previous part to find all theeigenvalues of T and describe their eigenvectors. Check that thefunctions you found are indeed eigenvectors of T.
