Name:      

Directions: Show all work. No credit for answers without work.

1. [2 parts, 2 points each] Let $A=\left[\begin{array}{cc}\hfill 1& \hfill 0\\ \hfill 0& \hfill -1\end{array}\right]$ and let ${𝐱}_0=\left[\begin{array}{c}\hfill 1\\ \hfill 1\end{array}\right]$.

   1. Apply the power method to compute ${𝐱}_k$ and ${\mu }_k$ for $0\le k\le 3$.
   2. Note that ${𝐱}_k$ is not approaching the direction of an eigenvector of $A$. Why does this not contradict the power method?

2. [4 points] Given $𝐲$ and $𝐯$ below, decompose $𝐲$ as $𝐲=c𝐯+𝐳$ where $c$ is a scalar and $𝐳\cdot 𝐯=0$.

   $$\begin{array}{llllllll}\hfill 𝐲& =\left[\begin{array}{c}\hfill 2\\ \hfill 2\\ \hfill 1\end{array}\right]& \hfill 𝐯& =\left[\begin{array}{c}\hfill -3\\ \hfill 1\\ \hfill -2\end{array}\right]& \hfill & & \hfill & \end{array}$$

3. [2 points] Let $W=\mathrm{Span}\{{𝐯}_1,\dots ,{𝐯}_p\}$. Prove that if $𝐳\cdot {𝐯}_i=0$ for $1\le i\le p$, then $z\in W^{\perp }$.