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We will start this chapter off by looking at the application
of matrices that almost every book on Linear Algebra starts off with, solving
systems of linear equations. Looking at
systems of equations will allow us to start getting used to the notation and
some of the basic manipulations of matrices that we’ll be using often
throughout these notes.
Once we’ve looked at solving systems of linear equations
we’ll move into the basic arithmetic of matrices and basic matrix
properties. We’ll also take a look at a
couple of other ideas about matrices that have some nice applications to the
solution to systems of equations.
One word of warning about this chapter, and in fact about
this complete set of notes for that matter, we’ll start out in the first
section or to doing a lot of the details in the problems, but towards the end
of this chapter and into the remaining chapters we will leave many of the
details to you to check. We start off by
doing lots of details to make sure you are comfortable working with matrices
and the various operations involving them.
However, we will eventually assume that you’ve become comfortable with
the details and can check them on your own.
At that point we will quit showing many of the details.
Here is a listing of the topics in this chapter.
Systems of Equations 
In this section we’ll introduce most of the
basic topics that we’ll need in order to solve systems of equations including
augmented matrices and row operations.
Solving Systems of Equations Here we will look at the Gaussian Elimination
and Gauss-Jordan Method of solving systems of equations.
Matrices
We will introduce many of the basic ideas and
properties involved in the study of matrices.
Matrix Arithmetic & Operations In this section we’ll take a look at matrix
addition, subtraction and multiplication.
We’ll also take a quick look at the transpose and trace of a matrix.
Properties of Matrix Arithmetic We will take a more in depth look at many of
the properties of matrix arithmetic and the transpose.
Inverse Matrices and Elementary Matrices Here we’ll define the inverse and take a look
at some of its properties. We’ll also
introduce the idea of Elementary Matrices.
Finding Inverse Matrices In this section we’ll develop a method for
finding inverse matrices.
Special Matrices We will introduce Diagonal, Triangular and
Symmetric matrices in this section.
LU-Decompositions In this section we’ll introduce the
LU-Decomposition a way of “factoring” certain kinds of matrices.
Systems Revisited Here we will revisit solving systems of
equations. We will take a look at how
inverse matrices and LU-Decompositions can help with the solution process. We’ll also take a look at a couple of other
ideas in the solution of systems of equations.