%!TEX TS-program = latex
\documentclass[pdf,slideColor,colorBG]{prosper}
%style options: frames, autumn, rico, azure, troispoints, gyom, nuancegris, darkblue, contemporain, alienglow
\usepackage{times}

\usepackage{amssymb}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{examples}[example]{Example Set}

\title{ Intro to Linear Algebra \\ \ MAT 5230, \S 101\ }
\author{Wm C Bauldry \\ Autumn Semester, 2005}

\newcounter{e_temp}
\newcommand{\jmp}{\centerline{%
	\begin{tabular}{*{3}{p{0.3\linewidth}}}
	& \hfill \href{javascript:history.go(-1)}{\textcolor{blue}{\tiny{(Go Back)}}} \hfill \ 
	& \hfill \href{http://www.mathsci.appstate.edu/~wmcb/Class/5230/16_LinTrans_IV.tex}{\textcolor{blue}{\tiny{(View \LaTeX\ source)}}}
	\end{tabular}}}

\begin{document}
\graphicspath{{Images/}}
%%%%%%
%\maketitle

%%%%%%
\begin{slide}{Transformation Spaces}
\begin{definition} For $S$ and $T$ in $L(X,Y)$ and $\alpha$ in $F,$ define
\begin{enumerate}
\item $S+T$ by $(S+T)(x) \stackrel{\Delta}{=} S(x)+T(x)$
\item $\alpha\,S$ by $(\alpha\,S)(x)\stackrel{\Delta}{=} \alpha\,S(x)$ 
\item $S T$ by $(S T)(x) \stackrel{\Delta}{=} S\left(T(x)\right)$ when $\mathrm{range}(T)\subseteq\mathrm{dom}(S)$ \\[3ex]
\end{enumerate}
\end{definition} 
\begin{theorem} Let $S,T,U\in L(X,X).$ Then 
\begin{enumerate}
\item If $ST=US=I,$ then $S$ is bijective and $S^{-1}=T=U.$
\item If $S$ is bijective, then $(S^{-1})^{-1}=S.$
\item If $S$ and $T$ are bijective, then $(ST)^{-1}=T^{-1}S^{-1}.$
\item If $S$ is bijective and $\alpha\ne 0,$ then $(\alpha S)^{-1} = (1/\alpha)\cdot S^{-1}.$
\end{enumerate}
\end{theorem}
\end{slide}

%%%%%%
\begin{slide}{Polynomials of Transforms} 
\begin{theorem} $L(X,X)$ is an associative algebra%
\footnote{ ``Vector space plus multiplication.'' See pg.~56 and 104 of the text.}
with identity over $F$ (using 1, 2, \& 3, and identity $I(x)=x$). $L(X,X)$ is usually noncommutative. \\[1.5ex]
\end{theorem}
\begin{definition}[Powers of Transforms] Let $T\in L(X,X).$ Then set $T^{0}=I$ and, for $n>0$, define
$T^{(n)} \stackrel{_\Delta}{=} T\cdot T^{(n-1)}$ and $T^{(-n)}\stackrel{_\Delta}{=} \left(T^{-1}\right)^n.$ \\[1ex]
\end{definition}
\begin{definition} Let $p\in\mathbb{P}^n,$ so that $p(\lambda)=a_0+a_1 \lambda+\cdots + a_n \lambda^n.$ For $T\in L(X,X),$ define \\[-1.25ex]
\[  p(T) = a_0\, I+a_1 T+\cdots + a_n T^n = \sum_{i=0}^n \alpha_i T^i.  \]
\end{definition}
\end{slide}

%%%%%%
\begin{slide}{Finite Dimension Structure Theorem} 
\begin{definition} $X$ is \emph{isomorphic} to $Y,$ written $X\cong Y,$ if and only if there is a bijection $T\in L(X,Y).$ \\[2ex]
\end{definition}
\begin{theorem}[Structure Theorem] Every $n$-dimensional vector space $X$ over the field $F$ is isomorphic to $F^n.$ \\[1ex]
\end{theorem}
\textbf{Pf}. Choose a basis $\mathcal{B}=\{e_1, \dots, e_n\}$ for $X$. Then define $T\in L(X,F^n)$ by \\[-1.5ex]
\[  T\left(\sum_{i=1}^n \alpha_i e_i \right) = [\alpha_1, \alpha_2, \dots, \alpha_n]  \]
\begin{corollary}
Let $F$ be a field and $n$ be a positive integer. There is exactly one vector space of dimension $n$ over $F.$ \\[2ex]
\end{corollary}
\jmp
\end{slide}


\end{document}
\end

%%%%%%
\begin{slide}{} 
\end{slide}

\centerline{%
	\begin{tabular}{*{3}{p{0.3\linewidth}}}
	& \hfill \href{javascript:history.go(-1)}{\textcolor{blue}{\tiny{(Go Back)}}} \hfill \ 
	& \hfill \href{http://www.mathsci.appstate.edu/~wmcb/Class/5230/16_LinTrans_IV.tex}{\textcolor{blue}{\tiny{(View \LaTeX\ source)}}}
	\end{tabular}}
