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\headers{Math343}{Homework 14}{Due Dec 10, 2025}

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\noindent
\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  Show all work; no credit for solutions without work.

\begin{enumerate}

\item {} [6.2.9] Define vectors as follows.
\begin{align*}
u_1 &= \mtrix{1}{1\\ 0 \\ 1} & u_2 &= \mtrix{1}{-1\\4\\1} & u_3 &= \mtrix{1}{2\\1\\-2} & x &= \mtrix{1}{8\\-4\\-3}
\end{align*}
\begin{enumerate}
	\item Show that $\{u_1,u_2,u_3\}$ is an orthogonal set.
	\item Express $x$ as a linear combination of $u_1$, $u_2$, and $u_3$.
\end{enumerate}

\item {} [6.2.14] Let $\vect{y} = \mtrix{1}{2\\6}$ and $\vect{u}=\mtrix{1}{7\\1}$.  Write $\vect{y}$ as the sum of two orthogonal vectors, one in $\Span\{\vect{u}\}$ and one orthogonal to $\vect{u}$.

\item {} [6.2.15] Let $\vect{y} = \mtrix{1}{3\\1}$ and $\vect{u}=\mtrix{1}{8\\6}$.  Compute the distance from $\vect{y}$ to the line through $\vect{u}$ and the origin.

\item {} [6.2.25] Let $U$ be an $m\times n$-matrix with orthonormal columns.  Prove that $(U\vect{x})\cdot (U\vect{y}) = \vect{x}\cdot\vect{y}$ for all $\vect{x},\vect{y}\in\RR^n$.  

\item {} [6.3.12] Find the closest point to $\vect{y}$ in the subspace $W$ spanned by $\vect{v}_1$ and $\vect{v}_2$, and then find the distance from $\vect{y}$ to $W$.
\begin{align*}
\vect{y} &= \mtrix{1}{3\\-1\\1\\13} & \vect{v}_1 &= \mtrix{1}{1\\-2\\-1\\2} & \vect{v}_2 &= \mtrix{1}{-4\\1\\0\\3}
\end{align*}

\item {} [6.3.21] True/False.  All vectors and subspaces are in $\RR^n$.  Justify each answer.
\begin{enumerate}
	\item If $\vect{z}$ is orthogonal to $\vect{u}_1$ and $\vect{u}_2$ and if $W=\Span\{\vect{u}_1,\vect{u}_2\}$, then $\vect{z}\in W^{\perp}$.
	\item For each $\vect{y}$ and each subspace $W$, the vector $\vect{y}-\proj_W \vect{y}$ is orthogonal to $W$.
	\item The orthogonal projection $\hat y$ of $y$ onto a subspace $W$ can sometimes depend on the orthogonal basis for $W$ used to compute $\hat y$.
	\item If $\vect{y}$ is in a subspace $W$, then the orthogonal projection of $\vect{y}$ onto $W$ is $\vect{y}$ itself.
	\item If the columns of an $n\times p$ matrix $U$ are orthonormal, then $UU^T\vect{y}$ is the orthonormal projection of $\vect{y}$ onto the column space of $U$.
\end{enumerate}


\end{enumerate}
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