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\begin{document}
\headers{Math343}{Quiz 13}{Nov 21, 2025}

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\noindent
\textbf{Directions:} Show all work.  No credit for answers without work.

\begin{enumerate}

\item\ppoints{2}{2} Let $A = \mtrix{2}{1 & 0\\0 & -1}$ and let $\vv x_0 = \mtrix{2}{1 \\ 1}$.  
\begin{enumerate}
	\item Apply the power method to compute $\vv x_k$ and $\mu_k$ for $0\le k\le 3$.
	\vfill
	
	\item Note that $\vv x_k$ is not approaching the direction of an eigenvector of $A$.  Why does this not contradict the power method?
	\vfill
\end{enumerate}

\item\points{4} Given $\vect{y}$ and $\vect{v}$ below, decompose $\vect{y}$ as $\vect{y} = c\vect{\vect v} + \vect{z}$ where $c$ is a scalar and $\vect{z}\cdot\vect{v} = 0$.
\begin{align*}
\vect{y} &= \mtrix{1}{2\\2\\1} & \vect{v} &= \mtrix{1}{-3\\1\\-2}
\end{align*}
\vfill

\newpage
\item\points{2} Let $W=\Span\{\vv v_1,\ldots, \vv v_p\}$.  Prove that if $\vv z \cdot \vv v_i = 0$ for $1\le i\le p$, then $z\in W^{\perp}$.
\vspace{3cm}


\end{enumerate}
\end{document}
