
\section*{Preface}

These linear algebra lecture notes are designed to be presented as twenty five, fifty minute lectures
suitable for sophomores likely to use the material for applications but still requiring
a solid foundation in this fundamental branch of mathematics. The main idea of the course is
to emphasize the concepts of vector spaces and linear transformations as mathematical
structures that can be used to model the world around us. Once ``persuaded'' of this truth, students
learn explicit skills such as Gaussian elimination and diagonalization in order that vectors and linear transformations
become calculational tools, rather than abstract mathematics.

In practical terms, the course aims to produce students who can perform computations with large linear systems
while at the same time understand the concepts behind these techniques. Often-times when a problem can be
reduced to one of linear algebra it is ``solved''. These notes 
do not devote much space to applications (there are already a plethora of textbooks with titles involving some
permutation of the words ``linear'', ``algebra'' and ``applications''). Instead, they attempt to explain the fundamental
concepts carefully enough that students will realize for their own selves when the particular application they encounter
in future studies is ripe for a solution via linear algebra.

The notes are designed to be used in conjunction with a set of online homework exercises which teach 
basic linear algebra skills. These are a set of nine Webwork assignments which are collected weekly
and available at

\begin{quote}
\href{
http://webwork.math.ucdavis.edu/}{http://webwork.math.ucdavis.edu/}
\end{quote}

\noindent
Webwork is an open source, online homework system which originated at the University of Rochester. It can efficiently check whether a student
has answered an explicit, typically computation-based, problem correctly. The problem sets chosen to accompany these
notes could contribute roughly a 20\% of a student's grade, and ensure that basic computational skills are mastered.
Most students rapidly realize that it is best to print out the Webwork assignments and solve them on paper before 
entering the answers online. Those who do not tend to fare poorly on midterm examinations. We have found that there 
tend to be relatively few questions from students in office hours about the webwork assignments. Instead, by assigning 20\%
of the grade to written assignments drawn from  problems chosen randomly from the review exercises at the end of each lecture, the
student's focus was primarily on understanding ideas. They range from simple tests of understanding of the material in the lectures
to more difficult problems, all of them require thinking, rather than blind application of mathematical ``recipes''. 
Office hour questions reflected this and offered an excellent chance
to give students tips how to present written answers in a way that would convince the person grading their work that they deserved full credit!

Each lecture concludes with references to the comprehensive online textbook of Jim Hefferon:

\begin{quote}
\href{http://joshua.smcvt.edu/linearalgebra/}{http://joshua.smcvt.edu/linearalgebra/}
\end{quote}
and the notes are also hyperlinked to Wikipedia where students can rapidly access further details and background material
for many of the concepts.
There are also an array of useful commercially available texts such as
\begin{itemize}
\item ``Introductory Linear Algebra, An Applied First Course'', B. Kolman and D. Hill, Pearson 2001.
\item ``Algebra and Geometry'', D. Holten and J. Lloyd, CBRC, 1978.
\item ``Theory and Problems of Linear Algebra'', S. Lipschutz, McGraw-Hill 1987.
\end{itemize}

There are still many errors in the notes, as well as awkwardly explained concepts. An army of 200 students have already found
many of them. The review exercises would provide a better survey of what linear algebra really is if there were more ``applied''
questions. We welcome your contributions!

\begin{quote}
Andrew and Tom.
\end{quote}

\vspace{4cm}
\noindent
\copyright 2009 by the authors. These lecture notes may be reproduced in their entirety for non-commercial purposes.

\newpage
