###### 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*}
1. Find the eigenvalues of $$A$$ and find a basis for the associated eigenspaces.

2. 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.

3. Write $$A\xvec$$ as a linear combination of basis vectors.

4. What is $$\coords{\xvec}{\bcal}\text{,}$$ the representation of $$\xvec$$ in the coordinate system defined by $$\bcal\text{?}$$

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

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

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