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\title{MA20008 Algebra 1, 2004, Sheet 6}
\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
${\bf w_1}, {\bf w_2}$. Suppsoe 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 $W$.
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 $A$.
\end{enumerate}
\end{document}