\documentclass[12pt]{article}
\usepackage{latexsym}
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\title{Algebra 1; MA20008; Sheet 2}
\author{{\tt G.C.Smith@bath.ac.uk}}
\date{18-x-2004}
\begin{document}
\maketitle
\begin{enumerate}
\item Suppose that $U$ and $V$ are vector spaces over the
same field $F$, and that
$W$ is a subspace of $V$. Let $\alpha: U \rightarrow V$ 
be a linear map. Show that 
$Y = \{ {\bf x } \mid {\bf x } \in U,\ \alpha({\bf x}) \in V\}$
is a subspace of $U$.
\item Suppose that $U$ and $V$ are vector spaces over the same
field $F$, and
that we have linear maps $\alpha:  U \longrightarrow V$
and $\beta:  U \longrightarrow V$. 
Show that $Z = \{  {\bf x } \mid {\bf x } \in U,\ \alpha({\bf x })
= \beta({\bf x }) \}$ is a subspace of $U$.
\item Let $\mathbb C$ be the complex numbers, viewed 
as a vector space over $\mathbb R$. We have shown that
the map $\varphi : \mathbb C  \longrightarrow \mathbb C$
defined by complex conjugation is a linear map. Let
$n$ be a natural number, and define $\theta_n$ to
be multiplication by $e^{\frac{2\pi i}{n}}$; more formally
$\theta_n : {\mathbb C} \longrightarrow {\mathbb C}$ with
$\theta_n(z) = e^{\frac{2\pi i}{n}}z$ for all $z \in {\mathbb C}.$
\begin{enumerate}
\item[(a)] Show that each map $\theta_n$ is linear.
\item[(b)] How many different maps can you get by composing
the maps $\theta_4$ and $\theta_6$? {\em (For example, 
$\theta_4 \theta_4 \theta_6 \theta_6 \theta_4$ is one such
composition.)}
\item[(c)] How many different maps can you get by composing
the maps $\theta_5$ and $\varphi$?
\item[(d)] How many different maps can you get by composing
the maps $\theta_4$, $\theta_6$ and $\varphi$?
\end{enumerate}
\item Let $V$ we a vector space over a field $F$. We define 
a {\em line} as follows. Suppose that ${\bf a }, {\bf b} \in V$
with ${\bf b } \not = {\bf 0}.$ The set
\[ L = \{ {\bf r} \mid {\bf } {\bf r}  = {\bf a} + t{\bf b},\ t \in F\}\] 
is a {\em line}. Suppose that $U$ is also a vector space over $F$
and that \[\alpha : V \longrightarrow U\] is a linear map.
Show that if ${\bf b} \not \in \mbox{ Ker }\alpha$,
then \[K = \{ \alpha({\bf r}) \mid  {\bf r} \in L\}\] is a line.
What happens if ${\bf b } = {\bf 0}?$  
\item Regard ${\mathbb R}^n$ as a vector space over $\mathbb R$.
Define a map $\mu : {\mathbb R}^n \longrightarrow {\mathbb R}^n$
by $(x_1, x_2, \ldots, x_n) \mapsto (x_2, x_3, \ldots, x_n,0)$
for all $(x_1, x_2, \ldots, x_n) \in \mathbb R^n.$
\begin{enumerate}
\item[(a)] Show that $\mu$ is a linear map.
\item[(b)] Show that $\mu^n$ is the zero map ($\mu^n$ denotes
the map obtained by composing $n$ copies of $\mu$).
\item[(c)] Show that $\mu^{n-1}$ is not the zero map.
\end{enumerate}
\item Let $V$ be a vector spaces, and suppose that
$\alpha$ and $\beta$ are both projections onto subspaces
of $V$ with suitable kernels. Suppose also that 
$\alpha \beta= \beta \alpha$. Show that $\alpha \beta$
is a projection. 
\end{enumerate}
\end{document}
