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{\Large {Mathematics 3A, Spring 1998}}\\
\large Review Guide for Final Exam
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Computational Techniques:
\begin{enumerate}

\item Solve a system of linear equations.
\item Invert a matrix.
\item Find a basis for a subspace of a vector space (by using
coordinates); determine if a given set of vectors is linearly
independent.
\item Compute a determinant.
\item Diagonalize a matrix (possibly to compute powers of that matrix,
or to solve a system of linear differential equations).
\item Find the matrix for a linear transformation with respect to a
given basis; change the representation to a different basis.  Note:
follow the strategy from the lectures, not from the book.
\item Orthogonal projections (Wednesday, June 10 lecture).

\end{enumerate}

Theoretical Topics:
\begin{enumerate}

\item Determine if a given set $V$ together with an addition $\oplus$
and a scalar multiplication $\odot$ is a vector space (p.~189 no.~1,
3,~9).

\item Determine if a given subset $W$ of a vector space $V$ is a
subspace (p.~203 no.~27, 34,~36).

\item Determine if a given transformation is linear (p.~226 no.~1).

\item Determine if a given linear transformation is invertible.
Important: Know the relationship between linear transformations and
matrices, and the various invertibility criteria for matrices (p.~228,
no.~38, 49; is the transformation given in no.~31 invertible?).

\item Prove that some abstract set of vectors is linearly independent,
or is a basis (p.~136 no.~34, 36).

\item Properties of determinants (list from lecture!); things you can
prove with and about determinants (p.~262 no.~30, 31,~32).

\item Range, kernel, nullspace, nullity, rank, dimension, and how they
are related (p.~141 no.~18, 20,~23).

\item Properties of linear transformations (p.~153 no.~29, 31).

\item Properties of eigenvalues and eigenvectors (p. 310 no.~24, 25,
27, 30, 35, 41; p.~316 no.~22).

\end{enumerate}

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