%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Math 3130 
%  May 2013
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[11pt]{amsart}
\usepackage{amsmath,amsthm,amsfonts,latexsym,amssymb,url,color}

\pagestyle{headings}
\setlength{\textwidth}{468true pt}
\setlength{\headheight}{0true pt} % needed for 12pt documents
\setlength{\oddsidemargin}{0 truept}
\setlength{\evensidemargin}{0 truept}
%\setlength{\textheight}{580true pt}
\setlength{\textheight}{580true pt}

\begin{document}

\begin{center}
{\bf Math 3130, Introduction to Linear Algebra}
\end{center}

\bigskip
\bigskip


\noindent
\emph{Catalog description:}
Examines basic properties of systems of linear equations, vector spaces,
linear independence, dimension, linear transformations, matrices,
determinants, eigenvalues, and eigenvectors. 
Prereq., MATH 2300 or APPM 1360.\footnote{Calculus 2.} 
Credit not granted for this course and 
APPM 3310.\footnote{Matrix Methods.}
%, taught by Applied Math. 
%The catalogue description for this
%competing course is: ``Introduces linear algebra and matrices, with an
%emphasis on applications, including methods to solve systems of linear
%algebraic and linear ordinary differential equations. Discusses vector
%space concepts, decomposition theorems, and eigenvalue problems. Credit not
%granted for this course and MATH 3130. Prereq., APPM 2350 or 2360.'' 
%The goals of this course are described in the 2013 syllabus as
%``The course goals are to understand the basic concepts of 
%linear algebra and to be introduced to some aspects of 
%computational techniques used in matrix methods. To accomplish
%these goals, we will study systems of linear equations 
%and the LU decomposition, vector spaces
%and subspaces, orthogonality, linear transformations 
%and the fundamental subspaces of a matrix,
%and eigenvalues and eigenvectors. 
%Matrix decompositions (e.g. LU, QR, SVD, etc.) will play a
%fundamental role throughout the course.''}

\bigskip

Math 3130, Introduction to Linear Algebra, 
is a course designed to prepare students for a wide range of
%succeeding 
courses in mathematics, physics, economics, computer science
and engineering.
%The target audience typically has a 
%mixed\footnote{In one Spring 2013 section the distribution
%of majors was: Mathematics(14), 
%Physics(8), Economics(7), Computer Science(4), Aerospace
%Engineering(3), Chemistry(3)
%Astronomy(2),
%Secondary Education-Mathematics(2),
%Mechanical Engineering(2),
%Generic Education Transfer(2),
%Applied Math(1),
%English(1),
%Film Studies(1),
%Integrative Physiology(1),
%Music(1),
%Philosophy(1).}
%background. Some will have taken a proof-oriented
%course before, some will not; some will have 
%seen complex numbers before, some will not.
%All are expected to have passed Calculus 2 before enrolling in Math 3130.
The course is taught with a mixture of theory 
and application. Students are expected to leave the course able to 
solve problems using linear algebra, 
to know a number of applications of linear algebra, 
to be able to follow complex logical arguments 
and to develop simple logical arguments. 

The chief theoretical topics and their objectives are:

\begin{enumerate}
\item
Systems of linear equations.
\begin{enumerate}
\item 
Set up and solve a system of linear equations using Gaussian
elimination.
\item 
Be able to distinguish between consistent and inconsistent
systems.
\item 
Relate the solutions of a system to the 
solutions of the associated homogeneous system.
\item
Rewrite a linear system as a matrix equation or a vector equation.
\end{enumerate}

\item
Matrices and matrix algebra
\begin{enumerate}
\item Be able to identity laws of matrix algebra,
and to give examples of matrices satisfying
$AB\neq BA$.
\item 
Explain the laws of transpose. Define
symmetric and antisymmetric matrices.
\item 
Explain left, right and 2-sided invertibility.
Be able to link each invertibility property for $A$
to the corresponding 
solvability property of the system $A{\bf x}={\bf b}$.
\item 
Define \emph{elementary matrices} and explain their
role in Gaussian elimination.
\item
Define \emph{(block) diagonal} and \emph{(block) triangular}
matrices.
\end{enumerate}

\item
Vector spaces
\begin{enumerate}
\item 
Explain the defining properties of vectors spaces and 
give examples of vector spaces.
\item 
Explain the geometric interpretation of the vector
space operations in $\mathbb R^n$.
\item 
Define the four fundamental subspaces associated
with a matrix.
\item 
Define \emph{linear combination}, \emph{linearly independent}, 
\emph{span}, 
\emph{basis}, and
\emph{dimension}.
\item 
Explain how to write a vector in coordinates
relative to a basis.
\item 
Compute the dimension of any of the four fundamental
subspaces.
\item 
Define \emph{rank} and \emph{nullity} and
prove the rank+nullity theorem.
\end{enumerate}


\item
Linear transformations
\begin{enumerate}
\item 
Define \emph{linear transformation}.
\item 
Explain why any linear transformation between 
finitely generated spaces has a matrix representation.
\item 
Compute the matrix of a transformation relative
to specified input and output bases.
\item 
Define \emph{image} and \emph{kernel}, and use these
words in definitions of 
\emph{onto} and \emph{1-1} transformations.
Explain the link between (image, kernel) and
(rank, nullity). Define \emph{isomorphism}.
\item 
Explain why any finite dimensional real vector space
is isomorphic to $\mathbb R^n$ for some $n$.
\end{enumerate}


\item
Determinant
\begin{enumerate}
\item 
Explain the connection between the determinant
and area in $\mathbb R^2$ and volume in $\mathbb R^3$.
\item 
Describe the cofactor expansion of the determinant
and be able to use it to compute some determinants.
\item 
Be able to compute the inverse of a matrix via the
determinant formula for inverse.
\end{enumerate}


\item
Eigenvalues and eigenvectors
\begin{enumerate}
\item 
State the definitions of 
\emph{eigenvalue}, \emph{eigenvector}, and \emph{eigenspace}.
\item 
Given a square matrix, find its characteristic polynomial,
eigenvalues and bases for its eigenspaces.
\item 
State necessary and sufficient conditions for a matrix to be 
diagonalizable.
\item 
Find a diagonalizing matrix, if one exists.
\item 
Define similarity, and explain how the characteristic polynomial,
eigenvalues and eigenspaces of similar matrices are related.
\item
Explain how to raise a diagonalizable matrix to a large power,
and how to use this to solve linear recurrences.
\end{enumerate}


\item
Orthogonality
\begin{enumerate}
\item 
Explain the concept of orthogonality of vectors
and subspaces.
\item 
Explain the orthogonality relations among the four fundamental subspaces.
\item 
Given ${\bf u}$ and ${\bf v}$, compute $\|{\bf u}\|$,
the unit vector in the direction of ${\bf v}$, 
the angle between ${\bf u}$ and ${\bf v}$.
\item 
State the Cauchy-Schwarz Theorem.
\item 
Construct a basis for the orthogonal complement of a 
set of vectors in $\mathbb R^n$. 
\item 
Given a least-squares problem, derive the normal equations and solve.
Be able to apply this to fit a curve to data.
\item 
Be able to apply the Gram-Schmidt 
process to construct an orthonormal set of vectors.
\item
Given a vector ${\bf u}$ and a subspace $W$ of $\mathbb R^n$,
compute the orthogonal
projection of ${\bf u}$ onto $W$.
\item
Be able to define orthogonal transformations and to give examples.
\end{enumerate}
\end{enumerate}


\bigskip

There are many types of applications that could be covered in a 
semester course. Fitting polynomial curves to data,
network flow and electrical network problems are
good for illustrating the practicality of linear systems,
and for gaining practice with Gaussian elimination.
Discrete dynamical systems and Markov chains,
where one might look for a steady state vector,
could be used to lead into a discussion of eigenvectors.
Linear recurrence relations illustrate the 
usefulness of diagonalizing. Any application 
can be used to illustrate the method of least squares.


\end{document}