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\noindent
\parbox{2.25in}{\bf Math 5230}
\hfill {\Large \bf Homework \#8} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Nov. $18^\mathrm{th}$, 2022}

\vspace{0.2in}

\begin{description}
\item[\#1 It's the Law] Let $V$ be an inner product space and $\B x, \B y \in V$. Prove the {\it parallelogram law}: 
$$\| \B x+\B y\|^2 + \|\B x-\B y\|^2 = 2\|\B x\|^2+2\|\B y\|^2$$
Then draw a picture of a parallelogram in the plane to explain what this means geometrically. 

\item[\#2 Easy Calculating] Suppose $W = \mathrm{col}(B)$ where $B = \begin{bmatrix}
1 & 0 & 1 & 1 & 2 \\  0 & 1 & -1 & 0 & 1 \\  -1 & 1 & -2 & 1 & 1 \\  2 & 1 & 1 & 0 & 3 \\ 1 & 0 & 1 & 1 & 2 \end{bmatrix}$.

Maple code:\\
{\color{red}\verb# B := <<1,0,-1,2,1>|<0,1,1,1,0>|<1,-1,-2,1,1>|<1,0,1,0,1>|<2,1,1,3,2>>; #}

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Find a basis for $W$ and $W^\perp$.

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\item[\#3 Not-So-Easy Calculating] Let $P_3 = \{ at^3+bt^2+ct+d \;|\; a,b,c,d \in \mathbb{R} \}$ (as usual). Turn $P_3$ into a (real) inner product space by defining $\displaystyle \langle f,g \rangle = \int_0^1 f(t)g(t)\,dt$ for all $f,g \in P_3$. 

\begin{enumerate}[(a)]
\item Explain why this is an inner product (run through the axioms).
\item Recall that $\mathrm{std.} = \{ 1,t,t^2,t^3\}$ is the standard basis for $P_3$. This is {\it not} an orthogonal basis. 

Write down the matrix for our inner product relative to the standard basis.

\item Use the Gram-Schmidt process on $\mathrm{std.}$ to find an orthogonal basis $\alpha$ for $P_3$. 

\item Let $W= \mathrm{span} \{f,g,h\}$ where $f(t) = t^3+t^2$, $g(t)=t^2+t$, $h(t)=t+1$.

Find $\alpha$-coordinates for $f$, $g$, and $h$. In fact, let $A = \Big[\,[f]_\alpha\;[g]_\alpha\;[h]_\alpha\,\Big]$.

\item {}[Extra Credit:] Since $\alpha$ is orthogonal, we know that a basis for the null space of $A^T$ (we just need a transpose since we're keeping it real) gives us a basis for the orthogonal complement (more-or-less in coordinates). Find a basis for the null space of $A^T$ and then use that to find a basis for $W^\perp$.

{\it Note:} Since $\alpha$ is orthogonal (and not necessarily orthonormal) the translation back from the world of coordinates requires some care.
\end{enumerate}

{\color{red}\large Warning:} Theoretically this problem could be done by hand, but I {\it pity the fool} that does. I did my integrations and such in Maple. The answers aren't terrifying, but they're also not exactly simple.

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