
\documentclass[12pt]{amsart}

\usepackage{241}

\title{21-241 Matrices and Linear Transformations \\
       Summer 1 2012 \\
       Course notes \\
       June 13}
\author{Paul McKenney}


\begin{document}

\maketitle

\section{Warm-up}

What's the determinant of a diagonal matrix $\diag(d_1,\ldots,d_n)$?  I can think of three proofs.
\begin{enumerate}
  \item  Row-ops.  [Hard to describe formally.]
  \item  Problem 8 on HW4 and induction.
  \item  Write the determinant using permutations.
\end{enumerate}

\section{Orthonormal Bases}

\begin{defn*}
  If $x\in\CC^n$ and $\norm{x} = 1$, we call $x$ a \emph{unit vector}.  Any nonzero vector can be scaled to make a unit vector, and we call this \emph{normalizing} the vector.

  If $x_1,\ldots,x_k\in\CC^n$ are distinct, pairwise orthogonal, unit vectors then we call the set $\{x_1,\ldots,x_k\}$ \emph{orthonormal}.
\end{defn*}

Recall that if $x_1,\ldots,x_k$ are nonzero, distinct, and pairwise orthogonal, then $\{x_1,\ldots,x_k\}$ is linearly independent.  Hence $\{x_1,\ldots,x_k\}$ makes up a basis for its own span.  In this case we call $\{x_1,\ldots,x_k\}$ an \emph{orthogonal basis} for the subspace $S = \lspan\{x_1,\ldots,x_k\}$.  If $\{x_1,\ldots,x_k\}$ is orthonormal then we call it an \emph{orthonormal basis} for $S$.

\begin{thm}
  Suppose $S$ is a subspace of $\CC^n$, and $\{x_1,\ldots,x_k\}$ is an orthogonal basis for $S$.  Then for any $y\in S$,
  \[
    y = \frac{\brak{y}{x_1}}{\norm{x_1}^2} x_1 + \cdots + \frac{\brak{y}{x_k}}{\norm{x_k}^2} x_k
  \]
  In particular, if $\{x_1,\ldots,x_k\}$ is orthonormal, then for any $y\in S$ we have
  \[
    y = \brak{y}{x_1} x_1 + \cdots + \brak{y}{x_k} x_k
  \]
\end{thm}

\begin{example*}
  Show that $\{e_1,\ldots,e_n\}$ is an orthonormal basis for $\CC^n$.
\end{example*}

\begin{example*}
  Show that $\{\smat{1 \\ 1 \\},\smat{-1 \\ 1 \\}\}$ is an orthogonal basis for $\CC^2$ but is not orthonormal.  What's the ``normalized'' version?  What are the coordinates of $e_1$ and $e_2$ in this orthonormal basis?
\end{example*}

\begin{defn*}
  Suppose $S$ is a $k$-dimensional subspace of $\CC^n$, and $\{s_1,\ldots,s_k\}$ is an orthonormal basis for $S$.  The \emph{orthogonal projection} of a vector $x\in\CC^n$ onto $S$ is the vector
  \[
    \PP_S(x) = \sum_{i=1}^k \brak{x}{s_i} s_i
  \]
\end{defn*}

\begin{example*}
  Let $s = \frac{1}{\sqrt{2}}\smat{1 \\ 1 \\}$, $S = \lspan\{s\}$, and $X = \lspan\{e_1\}$.  What's $\PP_S(e_1)$? $\PP_X(s)$? $\PP_S(s)$? $\PP_X(e_1)$?  What about $\PP_S(\PP_X(s))$?
\end{example*}

\begin{example*}
  Find an orthonormal basis for the plane $P$ in $\RR^3$ described by $3x - 2y + z = 0$.  Find the projections of $e_1,e_2,e_3$ onto $P$.
\end{example*}

\begin{thm}
  \begin{enumerate}
    \item  $\PP_S : \CC^n\to \CC^n$ is well-defined and a linear transformation.
    \item  If $P_S$ is the $n\times n$ matrix which implements $\PP_S$, then $P_S$ is a projection matrix.
    \item  For all $x\in\CC^n$, $\PP_S(x)$ is the unique vector in $S$ whose distance to $x$ is smallest.
  \end{enumerate}
\end{thm}

\end{document}
