%\documentclass{beamer} \documentclass[handout]{beamer} %%% produces a handout - ready to print, no frames \setbeamertemplate{navigation symbols}{ \insertslidenavigationsymbol} %keeps or removes navigation symbols \setbeamercolor{navigation symbols dimmed}{fg=blue!75!black} %changes how the bottom \setbeamercolor{navigation symbols}{fg=white!80!white} %navigation symbols look \mode {\usetheme{boxes}} \setbeamertemplate{items}[square] \hypersetup{colorlinks=true,linkcolor=gray} \usepackage{amsbsy,amsfonts} \usepackage{amstext} \usepackage{latexsym,eucal,amsmath,amsthm} \usepackage{amssymb} \usepackage{graphicx} \usepackage{subfigure} \usepackage{hyperref} \newcommand{\blob}{\rule[.2ex]{.9ex}{.9ex} \ } \newcommand{\dd}{\rule[.2ex]{.6ex}{.6ex}} \newcommand{\R}{\mathbb{R}} \newcommand{\norm}[1]{\lVert#1\rVert} % norm \newcommand{\avg}[1]{\left< #1 \right>} % average \newcommand{\sabs}[1]{\left| #1 \right|} % smaller absolute value \newcommand{\abs}[1]{\bigl| #1 \bigr|} % absolute value %\newcommand{\sabs}[1]{\left| #1 \right|} \title{ Linear Transformations} \author{Silvius Klein} \date{} \vspace{2in} \institute{MA1202 Linear Algebra with Applications} \AtBeginSection[] % "Beamer, do the following at the start of every section" { \begin{frame} \frametitle{Outline} % make a frame titled "Outline" \tableofcontents[currentsection] % show TOC and highlight current section \end{frame} } \begin{document} % this prints title, author etc. info from above %\begin{frame} %\titlepage %\end{frame} \begin{frame} \frametitle{General linear transformations} \underline{Definition}: Let $V$, $W$ be two vector spaces. A function $T \colon V \to W$ is called a {\em linear transformation} from $V$ to $W$ if the following hold for all vectors $u, v$ in $V$ and for all scalars $k$. \begin{itemize} \item[(i)] $T (u + v) = T (u) + T (v)$ (additivity) \item[(ii)] $T (k u) = k T (u)$ (homogeneity) \end{itemize} \pause If $V$ and $W$ are the same, we call a linear transformation from $V$ to $V$ a {\em linear operator}. \pause \medskip \underline{Theorem}: A function $T \colon V \to W$ is a linear transformation if and only if for all vectors $v_1, v_2$ in $V$ and for all scalars $k_1, k_2$ we have $$T (k_1 \, v_1 + k_2 \, v_2) = k_1 \, T (v_1) + k_2 \, T (v_2)$$ \end{frame} \begin{frame} \frametitle{General linear transformations} \underline{Theorem} (basic properties of linear transformations): If $T$ is a linear transformation then \begin{itemize} \item[a)] $T (\vec{0}) = \vec{0}$ \item[b)] $T ( - v) = - T (v)$ \item[c)] $T (u - v) = T (u) - T (v)$ \end{itemize} \pause \medskip \underline{Theorem}: If $T \colon V \to W$ is a linear transformation, $S = \{ v_1, v_2, \ldots, v_n \}$ is a basis in $V$, then for any vector $v$ in $V$ we can evaluate $T (v)$ by $$T (v) = c_1 \, T (v_1) + c_2 \, T (v_2) + \ldots + c_n \, T (v_n)$$ where $v = c_1 \, v_1 + c_2 \, v_2 + \ldots + c_n \, v_n$. \end{frame} \begin{frame} \frametitle{Kernel and range of a linear transformation} \underline{Definition}: Let $T \colon V \to W$ is a linear transformation. \begin{itemize} \item The set of all vectors $v$ in $V$ for which $T (v) = \vec{0}$ is called the {\em kernel} of $T$. We denote the kernel of $T$ by $\ker (T)$. \item The set of all outputs (images) $T (v)$ of vectors in $V$ via the transformation $T$ is called the {\em range} of $T$. We denote the range of $T$ by $R (T)$. \end{itemize} \pause \medskip \underline{Theorem}: If $T \colon V \to W$ is a linear transformation, then $\ker (T)$ is a {\em subspace} of $V$, while $R (T)$ is a subspace of $W$. \pause \medskip \underline{Definition}: If $V$ and $W$ are {\em finite} dimensional vector spaces and $T \colon V \to W$ is a linear transformation, then we call \begin{itemize} \item $\dim \ker (T) = \text{ nullity of } T$ \item $\dim R (T) = \text{ rank of } T $ \end{itemize} \pause \medskip \underline{Theorem}: If $V$ and $W$ are finite dimensional vector spaces and $T \colon V \to W$ is a linear transformation, then $$\text{ rank } (T) + \text{ nullity } (T) = \dim (V)$$ \end{frame} \begin{frame} \frametitle{One-to-one and onto functions} \underline{Definition} (one-to-one function): A function $f \colon X \to Y$ is called {\em one-to-one} if to distinct inputs it assigns distinct outputs. More precisely, $f$ is 1-1 means: if $x_1 \neq x_2$ then $f (x_1) \neq f (x_2)$. This is logically equivalent to saying that if $f (x_1) = f (x_2)$ then $x_1 = x_2$. \pause \bigskip \underline{Definition} (onto function): A function $f \colon X \to Y$ is called {\em onto} if every element in $Y$ is an output of $f$. More precisely, $f$ is onto if for every $y$ in $Y$ there is at least one $x$ in $X$ such that $f (x) = y$. \pause \bigskip Linear transformations are functions, so being one-to-one or onto applies (makes sense) for them as well. \end{frame} \begin{frame} \frametitle{Isomorphism} \underline{Theorem}: A linear transformation $T \colon V \to W$ is one-to-one if and only if $\ker (T) = \{ \vec{0} \}$. \medskip \underline{Theorem}: Let $T \colon V \to V$ be a linear operator, where $V$ is a \alert{finite} dimensional vector space. The following statements are equivalent. \begin{itemize} \item[a)] $T$ is one-to-one \item[b)] $\ker (T) = \{ \vec{0} \}$ \item[c)] $T$ is onto. \end{itemize} \medskip \underline{Definition}: A linear transformation $T \colon V \to W$ which is one-to-one and onto is called an {\em isomorphism}. Two vector spaces $V$ and $W$ are called {\em isomorphic} if there is an isomorphism $T \colon V \to W$. \medskip \underline{Examples}: $P_{n-1}$ is isomorphic to $\R^n$. $M_{2 \times 2} (\R)$ is isomorphic to $\R^4$. \medskip \underline{Theorem}: Every $n$ dimensional vector space is isomorphic to $\R^n$. \end{frame} \begin{frame} \frametitle{The matrix for a linear transformation: definition} We have: \begin{itemize} \item[$\diamond$] An $n$-dimensional vector space $V$ with a basis $B = \{ u_1, u_2, \ldots, u_n \}$. \item[$\diamond$] An $m$-dimensional vector space $W$ with a basis $B'$. \item[$\diamond$] A linear transformation $T \colon V \to W$. \end{itemize} \smallskip \underline{Definition}: The {\em matrix for $T$ relative to the bases $B$ and $B'$} is the $m \times n$ matrix $[ T ]_{B', B}$ defined by $$ [ T ]_{B', B} = \big[ \, [ T (u_1) ]_{B'} \, | \, [ T (u_2) ]_{B'} \, | \ldots | \,[ T (u_n) ]_{B'} \big] $$ \medskip Relative to these bases, we can think of the linear transformation $T$ as simply the multiplication transformation from $\R^n$ to $\R^m$ by the matrix $[ T ]_{B', B}$. More precisely, we have the following relation: $$ \big[ T (x) \big]_{B'} = [ T ]_{B', B} \cdot [ x ]_B$$ \end{frame} \begin{frame} \frametitle{The matrix for a linear transformation: properties} In class we have learned the following. \medskip \underline{Theorem}: If $T \colon V \to V$ is a linear operator and if $B$ is a basis for $V$, then the following are equivalent: \begin{itemize} \item[(a)] $T$ is one-to-one. \item[(b)] $[ T ]_{B, B}$ is invertible. \end{itemize} Moreover, if these conditions hold, then $$ [ T^{- 1} ]_{B, B} = [ T ]_{B, B}^{- 1}$$ \medskip But in fact much more is true: given any linear transformation $T$ from a vector space $V$ to another vector space $W$, the matrix for $T$ in two chosen basis $[ T ]_{B', B}$ completely encodes whether the transformation is one-to-one, onto or an isomorphism. \end{frame} \begin{frame} \frametitle{The matrix for a linear transformation: more properties} Let $V$ be an $n$-dimensional vector space with a basis $B$ and let $W$ be an $m$-dimensonal vector space with a basis $B'$. Let $T \colon V \to W$ be a linear transformation. \underline{Theorem}: The following are equivalent: \begin{itemize} \item[(a)] $T$ is one-to-one. \item[(b)] The null space of $[ T ]_{B, B}$ is $\{ \vec{0} \}$. \item[(c)] ${\rm nullity} \, [ T ]_{B', B} = 0$. \end{itemize} \smallskip \underline{Theorem}: The following are equivalent: \begin{itemize} \item[(a)] $T$ is onto. \item[(b)] The column space of $[ T ]_{B, B}$ is $\R^m$. \item[(c)] ${\rm rank} \, [ T ]_{B', B} = \dim (W)$. \end{itemize} \smallskip \underline{Theorem}: The following are equivalent: \begin{itemize} \item[(a)] $T$ is an isomorphism \item[(b)] $\dim (V) = \dim(W)$ and $[ T ]_{B', B}$ is invertible. \end{itemize} \end{frame} \end{document}