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\Large\bf
Linear Algebra in SAGE: Tutorial
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\begin{enumerate}
\item Matrices:
\begin{enumerate}
  \item {\tt matrix} command, {\tt random\_matrix} command, via a {\tt MatrixSpace}
  \item Compute the following associated to $A$: kernel, inverse (if invertible), characteristic
polynomial, eigenspaces (and understand the output), the product $A A$, echelon form,
Hessenberg form, determinant.
\item Create matrices form given rows or columns of $A$.  Make $A$ immutable.
\end{enumerate}
\item Vector spaces:
\begin{enumerate}
\item Choose a field $k$.  Create the $4$-dimensional vector space $V = k^4$ over $k$.
\item Create two three-dimensional subspaces $W_1$ and $W_2$ of $V$.
\item Compute the intersection of $W_1$ and $W_2$.  Compute their sum.
\item Express an element of $W_1 \cap W_2$ in terms of the basis for $W_1$ (using the
coordinates method).
\item Do the above 3 steps with $k$ replaced by $\Z$.
\end{enumerate}
\end{enumerate}