Exercise 8
Consider the matrix \(A=\left[\begin{array}{rr} 0.4 \amp 0.3 \\ 0.6 \amp 0.7 \\ \end{array}\right]\)
Find the eigenvalues of \(A\) and a basis for their associated eigenspaces.
Suppose that \(\xvec_0=\twovec{0}{1}\text{.}\) Express \(\xvec_0\) as a linear combination of eigenvectors of \(A\text{.}\)

Define the vectors
\begin{equation*} \begin{aligned} \xvec_1 \amp {}={} A\xvec_0 \\ \xvec_2 \amp {}={} A\xvec_1 = A^2\xvec_0 \\ \xvec_3 \amp {}={} A\xvec_2 = A^3\xvec_0 \\ \vdots \amp {}={} \vdots \end{aligned}\text{.} \end{equation*}Write \(\xvec_1\text{,}\) \(\xvec_2\text{,}\) and \(\xvec_3\) as a linear combination of eigenvectors of \(A\text{.}\)
What happens to \(\xvec_k\) as \(k\) grows larger and larger?