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{\Large {Mathematics 3A, Spring 1997}}\\
\large Review Guide for Midterm I
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\begin{enumerate}

\item Vector spaces (Section 3.1): Definition and basic properties
(working knowledge---you don't need to memorize the whole list of
axioms and theorems, but you should have a general idea and be able to
work with them).  Typical questions: Is an explicitly given $(X,
\oplus, \odot)$ a vector space?  (There are plenty of examples in the
current homework assignment.)  Not for this exam, but for the Final:
Prove (or disprove) some simple property (you should be able to
\emph{prove} or disprove any of the statements of p.~189 no.~18, and
of Theorem~3.1).

\item Linear combination, span (Section 3.2): Know \emph{precise}
definitions (see your lecture notes for a good way of writing these).
Typical question: $V$ is a vector space, $b \in V$, $X \subset V$ are
given.  Is ${b \in \Span X}$?  (See examples from current
homework.)  Be able to prove or disprove any property of p.~202
no.~25.

\item Linear dependence/independence for a set vectors, subspace,
basis, dimension (Section~3.2): Know \emph{precise} definitions as
given in lecture.  Typical questions: $V$ is a vector space, $X
\subset V$ is given.  Is $X$ linearly independent?  Is $X$ a subspace
of $V$?  Is $X$ a basis of $V$? (Again, see examples from this week's
homework.)

\item Coordinazation of vectors (Section~3.3):  One of the most
important concepts in L.A., it will be on the exam!  See p.~208,
Examples~3 and~4 (don't worry about Example~5 before the exam).  Do
the homework problems p.~202 no.~11, 19, 21, 23 by using
coordinazation techniques.

\item Solving a system of linear equations by Gauss--Jordan
elimination: Use the method taught in class (other methods may not
receive full credit).  Also: Know conditions for obtaining no
solution, a unique solution, or many solutions.

\item Given a matrix $A$, be able to define and compute the following:
a basis of its nullspace, a basis of its column space, a basis of its
row space, $\rank A$, $\nullity A$, and $A^{-1}$ if it exists.

Know the conditions for a matrix inverse to exist (the theorem with
six equivalent statements from the lecture---also see the theorem on
the handout, is a reduced version of the one given in class).  Equally
important is the \emph{rank equation} for a matrix $A \in M(m \times
n)$:
\[ \rank A + \nullity A = n. \]

\item Linear transformations (Section 2.3): Know \emph{precise}
definition, be able to check if a given transformation is linear, find
the matrix representation for a given transformation.  Very important:
Know that composition of transformations is equivalent to
multiplication of the corresponding matrices, and inversion of
transformations is equivalent to inversion of the corresponding
matrices. Be able to prove or disprove any of the statements of p.~153
no.~29.

\end{enumerate}

General notes:

\begin{itemize}

\item No multiple-choice or true/false questions.  You should be able
to either prove a statement, or disprove it by giving a counter
example.

\item No calculators.

\item Please arrive in time (i.e.\ 5 minutes early).

\item Remember the \emph{Give Up} option.

\end{itemize}

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