\documentclass[12pt]{amsart}
\usepackage{amssymb}
\usepackage{epsfig}
\input{shortcuts.tex}

\pagestyle{empty}
\setlength{\textwidth}{6.5in} 
\setlength{\textheight}{9in} 
\setlength{\headheight}{0in} 
\setlength{\evensidemargin}{0in}
\setlength{\oddsidemargin}{0in}
\setlength{\topmargin}{0in}


\newcommand{\F}{\bf F}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\va}{\mathbf{a}}
\newcommand{\vb}{\mathbf{b}}
\newcommand{\vc}{\mathbf{c}}
\newcommand{\ve}{\mathbf{e}}
\newcommand{\vu}{\mathbf{u}}
\newcommand{\vv}{\mathbf{v}}
\newcommand{\vw}{\mathbf{w}}
\newcommand{\vx}{\mathbf{x}}
\newcommand{\vy}{\mathbf{y}}
\newcommand{\vz}{\mathbf{z}}
\newcommand{\vzero}{\mathbf{0}}
\newcommand{\spn}{\operatorname{span}}
\newcommand{\poly}{\mathbb{P}}

\renewcommand{\labelenumi}{(\alph{enumi})}
\newcommand{\soln}[1]{
\vspace*{.1in}
\begin{quotation}
\noindent\emph{Solution.} #1
\end{quotation}
\vspace*{.1in}

}

\begin{document}
\begin{center}
{\bf Homework 4}\\
(due Wednesday, September 21)
\end{center}
\vspace*{.3in}

\noindent{\bf Book Problems:} 
\begin{description}
\item[1.2] 2.6
\item[p15] 3.1, 3.4
\end{description}

\begin{prob}
Let $V$ be a (non-trivial) vector space and $S\subset V$ be a finite generating set.  Show that $S$ contains a minimal generating set and that this is necessarily a basis for $V$.
\end{prob}

\begin{prob}
Let $f:X\to Y$ be a function between sets $X$ and $Y$.  If $A\subset X$, then we define the \emph{image} of $A$ under $f$ to be the set 
$$
f(A) := \{f(x)\in Y:x\in A\}.
$$
Similarly, if $B\subset Y$, then we define the \emph{preimage} of $B$ to be the set
$$
f^{-1}(B) := \{x\in X:f(x)\in B\}.
$$
Now let $T:V\to W$ be a linear transformation and decide whether each of the following statements is true or false.  If true, prove it.  If false, give a counterexample.
\begin{enumerate}
 \item If $H\subset V$ is a subspace, so is $T(H)$.  
 \item If $S = \{\vv_1,\dots,\vv_n\}\subset V$ is independent, so is $T(S)$.
 \item If $S\subset W$ is a finite set that generates $W$, then $T^{-1}(S)$ generates $W$.
 \item If $S\subset \{\vv_1,\dots,\vv_n\}\subset V$ is a set such that $T(S)$ is independent and no two vectors in $S$ have the same image under $T$, then $S$ is independent.
\end{enumerate}
\end{prob}

\begin{prob}
Following is a list of functions between various vector spaces (all over $\R$).  Determine, in each case, whether the function is a linear transformation.  If it isn't, give a \emph{specific} example illustrating failure of linearity.  If it is, justify \emph{one} (your choice, but mix it up a bit, for pete's sake) of the two conditions for linearity.  Note that I use $\cF(\R,\R)$ to denote the vector space of all functions $f:\R\to\R$, we let $P(\R)\subset \cF(\R,\R)$ be the subspace of all \emph{polynomial} functions, and we let $P_k(\R)$ denote the smaller subspace of polynomials with degree no larger than $k$.
\begin{enumerate}
\item $T:\R^3\to \R^3$, $T(x,y,z) = (x,x+y,x+y+z)$. 
\item $T:D\to\cF(\R,\R)$ ($D\subset \cF(\R,\R)$ is the subspace consisting of differentiable functions), $T(f(x)) = 2xf'(x)$.
\item $T:\cF(\R,\R) \to \R$, $T(f(x)) = f(5) + 2$.
\item $T:\R^2\to\R^2$, $T(x,y) = (|x|,|y|)$.
\item $T:P_2(\R)\to P_2(\R)$, $T(ax^2 + bx + c) = cx^2 + ax + b$.
\item $T:P(\R)\to P(\R)$, $T(p(x)) = p(x)^2$.
\item $T:\R^2\to\R$ by $T(\vw) = \vw\cdot (1,2)$.
\end{enumerate}
\end{prob}


\begin{prob}
Recall that $\C$ can be viewed as a real or a complex vector space (i.e. a
vector space over $\R$ or over $\C$).  We call a function $T:\C\to\C$
\emph{complex} linear if it is linear when $\C$ is regarded as a complex
vector space.  That is, $T(cz) = cT(z)$ for all \emph{complex} scalars
$c$.  Likewise $T$ is \emph{real} linear if it is linear when 
$\C$ is regarded as merely a real vector space.
\begin{enumerate}
\item Show that any complex linear transformation $T:\C\to\C$ has the form 
$T(z) = az$ for some $a\in\C$.  
\item Show that $T:\C\to\C$ given by $T(z) = \bar z$ is real linear but not
complex linear.
\end{enumerate}
\end{prob}

\begin{prob}
Show that if $T:U\to V$ and $S:V \to W$ are linear transformations, then so
is $S\circ T:U\to W$. 
\end{prob}

\begin{prob}
Let $V$ be a finite dimensional vector space and $W_1,W_2\subset V$ be subspaces.  Show that
$$
\dim (W_1 + W_2) = \dim W_1 + \dim W_2 - \dim W_1\cap W_2.
$$
(Hint: how could you get a useful basis for $W_1 + W_2$ by extending one for $W_1\cap W_2$? 
It takes some effort to get this problem just right.)
\end{prob}


\end{document}
\end

