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

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\begin{slide}{Singular and Nonsingular Examples}
\begin{examples} {\ }
\begin{itemize} 
\item Let $T([a,b]) = \left[\begin{array}{cc} 1 & 2 \\ 1 & 0 \end{array}\right] [a,b].$ Show $T$ is nonsingular.
\item Let $S([a,b]) = \left[\begin{array}{cc} 6 & 3 \\ 2 & 1 \end{array}\right] [a,b].$ Show $T$ is singular.
\item $\mathcal{D}:\mathbb{P}\rightarrow\mathbb{P}$ defined by $\mathcal{D}(p) = \displaystyle\frac{dp}{dx}$ is singular. \\[12pt]
\item Is $\mathcal{I}:\mathbb{P}\rightarrow\mathbb{P}$ defined by $\mathcal{I}(p) = \int p\, dx$ nonsingular?  \\[16pt]
\item Is $T([a,b]) = [a+b,0,a-b,0,0]$ invertible?
\end{itemize}
\end{examples} 
\end{slide}

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\begin{slide}{``Inverse Results''}
\begin{theorem}
Let $T\in L(X,Y)$ with $\mathrm{dim}(X)<\infty.$ Then $T$ is invertible if and only if $\rho(T) = \mathrm{dim}(X)$. $T$ is said to have ``full rank.'' \\[1ex]
\end{theorem}
\textbf{Pf}. \checkmark \\[1.5ex]

\begin{theorem}
Let $T\in L(X,Y)$ with $\mathrm{dim}(X)=\mathrm{dim}(Y)=n$ where $n<\infty.$ Then $T$ is invertible if and only if $\mathfrak{R}(T) = Y.$  \\[1ex]
\end{theorem}
\textbf{Pf}. $(\Rightarrow)$ $T$ invertible implies that $\mathrm{dim}(\mathfrak{R}(T))=n=\mathrm{dim}(Y).$ Since $\mathfrak{R}(T)$ is a subspace of $Y,$ then $\mathfrak{R}(T)=Y.$ \\[0.5ex]
$(\Leftarrow)$ Choose a basis $\mathcal{B}=\{y_1, \dots, y_n\}$ for $\mathfrak{R}(T)=Y.$ Then, since $T^{-1}(\mathcal{B})$ is an independent set of size $n,$ it forms a basis for $X$. Hence the only set of scalars for which $\sum_i \alpha_i x_i=0$ is $\alpha_i=0.$ Whence $\mathfrak{N}(T)=\{0\},$ so $T$ is invertible.
\end{slide}

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\begin{slide}{Collected Results, I}
{\footnotesize{
\begin{theorem}[Invertible Linear Transformations]
Let $X$ and $Y$ be vector spaces over $F$ and let $T\in L(X,Y)$. TFAE:
\begin{enumerate}
\item $T$ is invertible or nonsingular
\item $T$ is injective or $1$--$1$
\item $T(x)=0$ implies $x=0;$ i.e., $\mathfrak{N}(T) = \{0\}$
\item For each $y\in Y,$ $\exists$ a unique $x\in X$ such that $T(x)=y$
\item If $T(x_1)=T(x_2),$ then $x_1=x_2$
\item If $x_1\ne x_2,$ then $T(x_1)\ne T(x_2),$
\setcounter{e_temp}{\value{enumi}}
\end{enumerate}
If $X$ is finite dimensional, then TFAE:
\begin{enumerate}
\setcounter{enumi}{\value{e_temp}}
\item $T$ is injective
\item $\rho(T)=\mathrm{dim}(X)$
\end{enumerate}
\end{theorem}
}}
\end{slide}

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\begin{slide}{Collected Results, II}
\begin{theorem}[Surjective Linear Transformations]
Let $X$ and $Y$ be vector spaces over $F$ and let $T\in L(X,Y)$. TFAE:
\begin{enumerate}
\item $T$ is surjective or onto
\item For $y\in Y$, there is at least one $x\in X$ such that $T(x)=y$
\setcounter{e_temp}{\value{enumi}}
\end{enumerate}
If $X$ and $Y$ are finite dimensional, then TFAE:
\begin{enumerate}
\setcounter{enumi}{\value{e_temp}}
\item $T$ is surjective
\item $\rho(T)=\mathrm{dim}(Y)$ \\[3ex]
\end{enumerate}
\end{theorem}
\textbf{Pf}. \checkmark
\end{slide}

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\begin{slide}{Collected Results, III}
\begin{theorem}[Bijective Linear Transformations]
Let $X$ and $Y$ be vector spaces over $F$ and let $T\in L(X,Y)$. TFAE:
\begin{enumerate}
\item $T$ is bijective or onto
\item For $y\in Y$, there is a unique $x\in X$ such that $T(x)=y$
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\end{enumerate}
If $X$ and $Y$ are finite dimensional, then TFAE:
\begin{enumerate}
\setcounter{enumi}{\value{e_temp}}
\item $T$ is surjective
\item $\rho(T)=\mathrm{dim}(X)=\mathrm{dim}(Y)$  \\[2ex]
\end{enumerate}
\end{theorem}
\begin{theorem}[Common Finite Dimension]
Let $X$ and $Y$ be vector spaces over $F$ with finite dimension $n$ and $T\in L(X,Y)$. Then
\[  T\!:\!\mbox{injective} \Leftrightarrow  T\!:\!\mbox{surjective} \Leftrightarrow  T\!:\!\mbox{bijective} \Leftrightarrow  T\!:\!\mbox{invertible}  \]
\end{theorem}
\end{slide}

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\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 $S\circ T$ by $(S\circ T)(x) \stackrel{\Delta}{=} S\left(T(x)\right)$ \\[3ex]
\end{enumerate}
\end{definition} 
\begin{theorem} $L(X,Y)$ is a vector space over $F$ (using 1 \& 2) \\[4ex]
\end{theorem}
\begin{theorem} $L(X,X)$ is an associative algebra with identity over $F$ (using 1, 2, \& 3, and identity $I(x)=x$) 
\\[4.5ex]
\end{theorem}
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