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Suppose that we have a diagonalizable matrix \(A=PDP^{-1}\) where

Find the eigenvalues of \(A\) and find a basis for the associated eigenspaces.

Form a basis \(\bcal\) of \(\real^2\) consisting of eigenvectors of \(A\) and write the vector \(\xvec = \twovec{1}{4}\) as a linear combination of basis vectors.

Write \(A\xvec\) as a linear combination of basis vectors.

What is \(\coords{\xvec}{\bcal}\text{,}\) the representation of \(\xvec\) in the coordinate system defined by \(\bcal\text{?}\)

What is \(\coords{A\xvec}{\bcal}\text{,}\) the representation of \(A\xvec\) in the coordinate system defined by \(\bcal\text{?}\)

What is \(\coords{A^4\xvec}{\bcal}\text{,}\) the representation of \(A^4\xvec\) in the coordinate system defined by \(\bcal\text{?}\)