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\begin{document}
\centerline{Math 201 Homework for Friday, Week 13}
\bigskip
\textbf{Due:} Friday, December 3.

\begin{prob}
  Let $S=\{(1,0,i),(1,2,1)\}$ in $\C^3$ (with the standard inner product). Compute $S^\perp$.
\end{prob}

\begin{prob}
  Let~$V=\mathcal{P}(\R)$ be the vector space of all polynomials with coefficients in $\R$, with
    inner product~$\langle f(x),g(x)\rangle=\int_0^1f(t)g(t)\,dt$.  Let~$W=\mathcal{P}_1(\R)$. (Warning: to get this
    problem right, you will need to be very careful with your calculations and
  double-check your solutions.)
    \begin{enumerate}
      \item Find an orthonormal basis
        $\left\{ u_1,u_2 \right\}$ for~$W$.
      \item Find the closest polynomial in~$W$ to~$h(x)=3-2x+x^2$.  Express your solution
        in two forms: (i) as a linear combination of~$u_1$ and~$u_2$, and (ii) as
        a linear combination of~$1$ and~$x$.
    \end{enumerate}
\end{prob}

\begin{prob}
  Let $V$ be an $n$-dimensional vector space over $F=\R$ or $\C$, and let $\langle \, , \, \rangle$ be an inner product on $V$. Let $\mathcal{B}=\{ v_1, \dots, v_n \}$ be an ordered basis for $V$ (not necessarily orthonormal). Let $A$ be the $n\times n$ matrix given by
  \[A_{ij}=\langle v_j,v_i\rangle.\]
  Recall that for $x\in V$, $[x]_\mathcal{B}$ denotes the coordinate vector for $x$ with respect to the basis $\mathcal{B}$, and as usual, we will think of this vector in $F^n$ as an $n \times 1$ matrix.
  \begin{enumerate}
   \item Prove that for all $x,y\in V$, 
   \[\langle x,y \rangle = \left([y]_\mathcal{B}\right)^* A\left( [x]_\mathcal{B}\right).\]
   (Recall that for a matrix $C$,  we define~$\overline{C}$
     by~$\overline{C}_{ij}=\overline{(C_{ij})}$, and then we define the conjugate
   transpose by $C^\ast = \overline{C^t}$. \emph{Hint}: compute both sides using sum notation.  On the
   right-hand side, you will be computing the~$(1,1)$-entry of a~$1\times 1$ matrix.)
   \item Prove that the matrix $A$ satisfies $A=A^*$.
   \item If the basis $\mathcal{B}$ is orthonormal, what is the matrix $A$?
   \item (Bonus) Let $\mathcal{D}$ be another ordered basis for $V$, and let $C$ be the associated $n\times n$ matrix. How are $A$ and $C$ related?
  \end{enumerate}
\end{prob}

\end{document}


