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.

1. [6.2.9] Define vectors as follows.

   $$\begin{array}{llllllllllllllll}\hfill u_1& =\left[\begin{array}{c}\hfill 1\\ \hfill 0\\ \hfill 1\end{array}\right]& \hfill u_2& =\left[\begin{array}{c}\hfill -1\\ \hfill 4\\ \hfill 1\end{array}\right]& \hfill u_3& =\left[\begin{array}{c}\hfill 2\\ \hfill 1\\ \hfill -2\end{array}\right]& \hfill x& =\left[\begin{array}{c}\hfill 8\\ \hfill -4\\ \hfill -3\end{array}\right]& \hfill & & \hfill & & \hfill & & \hfill & \end{array}$$
   1. Show that $\{u_1,u_2,u_3\}$ is an orthogonal set.
   2. Express $x$ as a linear combination of $u_1$, $u_2$, and $u_3$.
2. [6.2.14] Let $𝐲=\left[\begin{array}{c}\hfill 2\\ \hfill 6\end{array}\right]$ and $𝐮=\left[\begin{array}{c}\hfill 7\\ \hfill 1\end{array}\right]$. Write $𝐲$ as the sum of two orthogonal vectors, one in $\mathrm{Span}\{𝐮\}$ and one orthogonal to $𝐮$.
3. [6.2.15] Let $𝐲=\left[\begin{array}{c}\hfill 3\\ \hfill 1\end{array}\right]$ and $𝐮=\left[\begin{array}{c}\hfill 8\\ \hfill 6\end{array}\right]$. Compute the distance from $𝐲$ to the line through $𝐮$ and the origin.
4. [6.2.25] Let $U$ be an $m\times n$-matrix with orthonormal columns. Prove that $(U𝐱)\cdot (U𝐲)=𝐱\cdot 𝐲$ for all $𝐱,𝐲\in {ℝ}^n$.

5. [6.3.12] Find the closest point to $𝐲$ in the subspace $W$ spanned by ${𝐯}_1$ and ${𝐯}_2$, and then find the distance from $𝐲$ to $W$.

   $$\begin{array}{llllllllllll}\hfill 𝐲& =\left[\begin{array}{c}\hfill 3\\ \hfill -1\\ \hfill 1\\ \hfill 13\end{array}\right]& \hfill {𝐯}_1& =\left[\begin{array}{c}\hfill 1\\ \hfill -2\\ \hfill -1\\ \hfill 2\end{array}\right]& \hfill {𝐯}_2& =\left[\begin{array}{c}\hfill -4\\ \hfill 1\\ \hfill 0\\ \hfill 3\end{array}\right]& \hfill & & \hfill & & \hfill & \end{array}$$

6. [6.3.21] True/False. All vectors and subspaces are in ${ℝ}^n$. Justify each answer.

   1. If $𝐳$ is orthogonal to ${𝐮}_1$ and ${𝐮}_2$ and if $W=\mathrm{Span}\{{𝐮}_1,{𝐮}_2\}$, then $𝐳\in W^{\perp }$.
   2. For each $𝐲$ and each subspace $W$, the vector $𝐲-{\mathrm{proj}}_W𝐲$ is orthogonal to $W$.
   3. The orthogonal projection $ŷ$ of $y$ onto a subspace $W$ can sometimes depend on the orthogonal basis for $W$ used to compute $ŷ$.
   4. If $𝐲$ is in a subspace $W$, then the orthogonal projection of $𝐲$ onto $W$ is $𝐲$ itself.
   5. If the columns of an $n\times p$ matrix $U$ are orthonormal, then $U{U}^T𝐲$ is the orthonormal projection of $𝐲$ onto the column space of $U$.