\documentclass[12pt]{article}
\usepackage{latexsym}
\usepackage{amssymb}
\title{Algebra 1; MA20008; Sheet 4 Solutions}
\author{{\tt G.C.Smith@bath.ac.uk}}
\date{2-xi-2004}
\begin{document}
\maketitle
\begin{enumerate}
\item Suppose that $n$ is a natural number or $0$,
and $F$ is a field. Show that there is a vector
space over $F$ of dimension $n$.
\item Suppose that $V$ is a vector space with subspaces
$U$ and $W$ both of dimension $n < \infty$.
Does it follow that $V$ is finite dimensional?
Does it follow that $V$ has dimension $n$?
Does it follow that $U = W$?
In each case you should supply a reason for your answer.
\item Let $V$ be a vector space of dimension $n$.
Suppose that $V_0, V_1, \ldots, V_m$ are subspaces
of $V$ with
\[ V_0 \leq V_1 \leq \cdots \leq V_m.\]
\begin{enumerate}
\item Suppose that $m > n$. Show that there 
is $i \in \{ 1,2, \ldots, m\}$ such that $V_i = V_{i-1}.$
\item Suppose that $m \leq n$. Show that it may be
that the spaces $V_0, V_1, \ldots, V_m$ are distinct.
\end{enumerate}
\item Suppose that $\alpha : U \longrightarrow W$
is a linear map between vector spaces over the same field.
Let ${\bf x_1}, {\bf x_2}, \ldots, {\bf x_n}$ be vectors in $U$.
\begin{enumerate}
\item  Suppose that $U = \langle {\bf x_1}, {\bf x_2}, \ldots, {\bf x_n} \rangle$ 
and $\alpha$
is surjective. Prove that $W = \langle \alpha({\bf x_1}), \alpha({\bf x_2}), 
\ldots, \alpha({\bf x_n}) \rangle.$
\item Suppose that ${\bf x_1}, {\bf x_2}, \ldots, {\bf x_n}$
are linearly independent and $\alpha$ is injective.
Show that $\alpha({\bf x_1}), \alpha({\bf x_2}),
\ldots, \alpha({\bf x_n}) \rangle$ are linearly independent.
\end{enumerate}
\newpage
\item Let $\zeta = e^{\frac{2\pi i}{5}} \in \mathbb C.$
\begin{enumerate}
\item Suppose that we view $\mathbb C$ as a vector space over
$\mathbb Q$. Show that $1, \zeta, \zeta^2, \zeta^3, \zeta^4$
are linearly independent.
\item Suppose that we view $\mathbb C$ as a vector space over
$\mathbb R$. Show that $1, \zeta, \zeta^2, \zeta^3, \zeta^4$
are linearly dependent.
\end{enumerate}
\item Let $V$ be a vector space with subspaces 
$U, W$ such that $U$ and $W$ are both finite 
dimensional. Let 
${\bf u_1}, {\bf u_2}, \ldots, {\bf u_m}$
be a basis of $U$ and 
${\bf w_1}, {\bf w_2}, \ldots, {\bf w_n}$ be
a basis of $W$.
\begin{enumerate}
\item Show that $U + W$ is finite dimensional. 
\item Show that ${\bf u_1}, {\bf u_2}, \ldots, {\bf u_m},
{\bf w_1}, {\bf w_2}, \ldots, {\bf w_n}$ need not be a basis
of $U + W$.
\item Suppose that $U + W = U \oplus W$. Show that
\[{\bf u_1}, {\bf u_2}, \ldots, {\bf u_m},
{\bf w_1}, {\bf w_2}, \ldots, {\bf w_n}\]
is a basis of $U+W$.
\item Suppose that ${\bf u_1}, {\bf u_2}, \ldots, {\bf u_m},
{\bf w_1}, {\bf w_2}, \ldots, {\bf w_n}$
is a basis of $U+W$. Show that $U+W = U \oplus W$. 
\end{enumerate}
\item Let $I$ be a set. Let $V$ be the set of real
valued functions on $I$; more formally
\[ V = \left\{ f \vert f: I \longrightarrow \mathbb R\right\}.\]
Define addition on $V$ by $(f + h)(x) :=
f(x) + g(x)$ for all $x \in I$. If $\lambda \in \mathbb R$
and $f \in V$ we define $\lambda \cdot f \in V$
by $(\lambda \cdot f)(x) = (\lambda)(f(x))$ where the final
multiplication is just the product (in $\mathbb R$).
\begin{enumerate}
\item Check that $V$ is now a vector space over $\mathbb R$.
\item For each $i \in I$, define a function $\delta_i \in V$
where $\delta_i(x) = \delta_{i,x}$ (Kr\"onecker delta).
Thus $\delta_i(i) = 1$ and $\delta_i(x) = 0$ if $x \not = i$.
Show that the vectors $\delta_i$ are linearly independent.
\item Let $W = \langle \delta_i : i \in I \rangle$ be the
span of all the $\delta_i$. Show that the vectors $\delta_i$
form a basis of $W$ (in that they are a linearly independent
spanning set for $W$).
\item Show that $W = V$ if and only if $I$ is finite.
\item Give an explicit example of a vector space with an 
clearly describable uncountable basis (no set theoretic
metaphysics allowed). 
\end{enumerate}
\end{enumerate}  
\end{document}