##### Preview Activity4.4.1

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

\begin{equation*} P = \left[\begin{array}{rr} 1 \amp -1 \\ 1 \amp 2 \\ \end{array}\right],\qquad D = \left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp -3 \\ \end{array}\right] \text{.} \end{equation*}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 \(\bcal\text{?}\)