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\title{MA20008 Algebra 1, 2004, Sheet 7}
\author{Geoff Smith, {\tt http://www.bath.ac.uk/$\sim$masgcs}}
\date{}
\begin{document}
\maketitle
\begin{enumerate}
\item Let $V$ be a two dimensional vector space over $\mathbb R$
with basis
${\bf v_1}, {\bf v_2}$. Let $W$ be a two dimensional vector space over $\mathbb R$
with basis
${\bf w_1}, {\bf w_2}$. Suppose that $\alpha: V \longrightarrow W$
is a linear map which has matrix
\[ \left(\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right)\]
with respect to these bases. Determine the matrix of $\alpha$ with
respect to new bases ${\bf v_1} + 2{\bf v_2}, {\bf v_1} - {\bf v_2}$ of $V$
and ${\bf w_1} + {\bf w_2}, {\bf w_1} - {\bf w_2}$ of $W$.
\item Suppose that $A$ is an $m \times n$ matrix with entries in a field
$F$. Show that there are vector spaces $V$ and $W$ over $F$ of dimensions $n$
and $m$ respectively, a linear transformation $\alpha: V \longrightarrow W$,
and choices of bases for these spaces so that the matrix of $\alpha$ with respect
to these bases is $A$.
\item Let $V$ and $W$ be vector spaces over $F$ of dimensions $n$
and $m$ respectively. Show that there is a choice of bases for 
$V$ and $W$ such that the matrix of $\alpha$ with respect to these bases
is \[ \left(\begin{array}{ll} I_r & 0_{r, n-r}\\ 0_{m-r, r} & 0_{m-r,n-r}\end{array}\right).\] 
\item Let $V = \mathbb R^2$ viewed as a vector space over $\mathbb R$, and
thought of as the Euclidean plane. Let $\rho_\theta: V \longrightarrow V$
denote the map which is rotation anticlockwise about the origin through
$\theta$ radians. Choose the standard basis ${\bf e_1}, {\bf e_2}$ for $V$,
where ${\bf e_1} = (1,0)$ and ${\bf e_2} = (0,1)$.
\begin{enumerate}
\item Show that $\rho_\alpha :  V \longrightarrow V$  is a linear map
for every $\alpha \in \mathbb R$.
\item Show that $\rho_\alpha \circ \rho_\beta = \rho_{\alpha + \beta}$ 
for all $\alpha, \beta \in \mathbb R$. 
\item Show that $\rho_\alpha$ and $\rho_\beta$ commute for all $\alpha, \beta \in \mathbb R$. 
\item Write down the matrix of $\rho_\alpha$ with respect to the given basis.
\item Deduce formulas for $\cos(\alpha + \beta)$ and $\sin(\alpha + \beta).$
\item Let $c$ denote complex conjugation. Calculate the matrix of 
$c \circ \rho_\theta \circ c$ and comment on what this means.
\end{enumerate}
\item Suppose that $A$ and $B$ are invertible $n \times n$ matrices with real entries.
Show that there are invertible $n \times n$ real matrices $D$ and $E$ such that
$DAE = B$.
\item Define a relation on $n \times n$ matrices with real entries
as follows: if $A, B \in \mbox{Mat}_{n,n}(\mathbb R),$ we write $A \sim B$
if and only if there are invertible $P, Q \in \mbox{Mat}_{n,n}(\mathbb R)$
such that $QAP = B$.
\begin{enumerate}
\item Show that $\sim$ is an equivalence relation.
\item How may equivalence classes are there?
\end{enumerate}
\end{enumerate}
\end{document}
