Changing-Bases-objectives.tex

\begin{enumerate}
\item Explain how to describe vectors using different coordinate systems. Be able to change between different coordinate systems, and use these ideas to construct matrix representations of linear transformations.
\item Show that the null space (kernel), column space (image), and eigenspaces of a matrix (linear transformation) are vector subspaces. Be able to extend a basis for these spaces to a basis for the domain or codomain.
\item For square matrices, explain why similar matrices $B=P^{-1}AP$ represent the same linear transformation (just using a different basis). Explain why the determinant, eigenvalues, rank, and nullity of a linear transformation are not dependent upon the basis chosen.
\item Explain how to diagonalize a matrix, as well as explain when it is or is not possible. 
\end{enumerate}