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. [5.8.8] Let $A=\left[\begin{array}{cc}\hfill 2& \hfill 1\\ \hfill 4& \hfill 5\end{array}\right]$ and let ${𝐱}_0=\left[\begin{array}{c}\hfill 1\\ \hfill 0\end{array}\right]$. Execute the power method to generate ${𝐱}_k$ and ${\mu }_k$ for $k=0,\dots ,4$, keeping 3 decimal places. What is the estimated eigenvalue/eigenvector pair?
2. Let $𝐮={\left[\begin{array}{cccc}\hfill 1& \hfill 2& \hfill 3& \hfill 4\end{array}\right]}^T$ and $𝐯={\left[\begin{array}{cccc}\hfill 1& \hfill -1& \hfill 2& \hfill -2\end{array}\right]}^T$. Find a scalar $\alpha$ and a vector $𝐰$ such that $𝐮=\alpha 𝐯+𝐰$ and $𝐰$ is orthogonal to $𝐯$.
3. [6.1.22] Let $𝐮={\left[\begin{array}{ccc}\hfill u_1& \hfill u_2& \hfill u_3\end{array}\right]}^T$. Explain why $𝐮\cdot 𝐮\ge 0$ directly from the definition of the dot product. When is $𝐮\cdot 𝐮=0$?
4. [6.1.24] Verify the parallelogram law for $𝐮$ and $𝐯$ in ${ℝ}^n$: $||𝐮+𝐯|{|}^2+||𝐮-𝐯|{|}^2=2||𝐮|{|}^2+2||𝐯|{|}^2$.

5. [6.1.19] True/False. Justify your answers.

   1. $𝐯\cdot 𝐯=||𝐯|{|}^2$
   2. For any scalar $c$, $𝐮\cdot (c𝐯)=c(𝐮\cdot 𝐯)$.
   3. If the distance from $𝐮$ to $𝐯$ equals the distance from $𝐮$ to $-𝐯$, then $𝐮$ and $𝐯$ are orthogonal.
   4. For a square matrix $A$, vectors in $\mathrm{Col}A$ are orthogonal to vectors in $\mathrm{Nul}A$.
   5. If vectors ${𝐯}_1,\dots ,{𝐯}_p$ span a subspace $W$ and if $𝐱$ is orthogonal to each ${𝐯}_j$ for $j\in \{1,\dots ,p\}$, then $𝐱$ is in $W^{\perp }$.