%% File: linAlgReviewGuide.tex
%%
%% Study Guide for Numerical Methods with MATLAB
%% Chapter 7, "A Review of Linear Algebra"
%%
%% Gerald Recktenwald, gerry@me.pdx.edu
%% August 2000
\documentclass{article}
\usepackage{studyGuide}
\renewcommand{\guideName}{Study Guide for Linear Algebra Review}
\title{Study Guide for Linear Algebra Review}
\author{Gerald Recktenwald}
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\begin{document}
%\maketitle
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\section*{Bare Essentials}
At the end of this chapter you should be able to
\begin{enumerate}
\item Identify which addition, subtraction, and multiplication operations
are legal for vectors and matrices.
\item Manually carry out legal addition, subtraction,
and multiplication operations involving vectors and matrices
\item Apply transpose operators to vectors and matrices.
\item Manually compute each of the vector norms, $L_1$, $L_2$, and $L_\infty$.
\item Compute and compare lengths of vectors with the $L_2$ norm.
\item Manually compute the inner product of two vectors.
\item Use an inner product to verify whether two vectors are orthogonal.
\item Use an inner product to verify whether two vectors are orthonormal.
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\end{enumerate}
\bigskip
\noindent To perform basic linear algebra with \MATLAB\ you will need to
\begin{enumerate}
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\item Define vector and matrix variables with assignment operations
\begin{itemize}
\item involving \texttt{[ \ldots ]}, and using spaces, commas,
and semicolons (as appropriate) to separate row and column
elements.
\item using built-in functions that return vectors and matrices.
\end{itemize}
\item Extract vector and matrix elements with subscripts.
\item Use colon notation to create ranges of subscripts, and thereby extract
or assign element values \emph{en masse}.
\item Use \MATLAB\ syntax to add, subtract, and multiply vectors and matrices.
\item Use the $L_2$ norm to write a \MATLAB\ expression to test whether two
vectors are ``equal to within a tolerance''.
\end{enumerate}
% -------------------------
\section*{An Expanded Core of Knowledge}
After mastering the bare essentials you should move on to a deeper understanding
of the fundamentals. Doing so involves being able to
\begin{enumerate}
\item Use row- and column-oriented algorithms for matrix--vector, vector--matrix,
and matrix--matrix multiplication.
\item Express conditions of linear dependence of vectors (row or column) in terms of
matrix rank.
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\end{enumerate}
\bigskip
\noindent To perform more advanced linear algebra operations with \MATLAB\ you will need to
\begin{enumerate}
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\item Express products of matrices with vectors and other matrices using
colon notation to make the row and column operations explicit.
\end{enumerate}
% -------------------------
\section*{Developing Mastery}
Working toward mastery of linear algebra you will need to
\begin{enumerate}
\item Integrate the linear algebra operations from Chapter~7 with the
material from Chapter~8 and Chapter~9. In particular, be able to
express row-wise operations in Gaussian elimination using colon notation.
\end{enumerate}
\end{document}