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Linear Algebra (Math 2318)
Here are my online notes for my Linear Algebra course that I
teach here at Lamar
University. Despite the fact that these are my “class
notes”, they should be accessible to anyone wanting to learn Linear Algebra or
needing a refresher.

These notes do assume that the reader has a good working
knowledge of basic Algebra. This set of
notes is fairly self contained but there is enough Algebra type problems
(arithmetic and occasionally solving equations) that can show up that not
having a good background in Algebra can cause the occasional problem.

Here are a couple of warnings to my students who may be here
to get a copy of what happened on a day that you missed.

- Because
I wanted to make this a fairly complete set of notes for anyone wanting to
learn Linear Algebra I have included some material that I do not usually
have time to cover in class and because this changes from semester to
semester it is not noted here. You
will need to find one of your fellow class mates to see if there is
something in these notes that wasn’t covered in class.

- In
general I try to work problems in class that are different from my
notes. However, with a Linear
Algebra course while I can make up the problems off the top of my head
there is no guarantee that they will work out nicely or the way I want
them to. So, because of that my
class work will tend to follow these notes fairly close as far as worked
problems go. With that being said I
will, on occasion, work problems off the top of my head when I can to
provide more examples than just those in my notes. Also, I often don’t have time in class
to work all of the problems in the notes and so you will find that some
sections contain problems that weren’t worked in class due to time restrictions.

- Sometimes
questions in class will lead down paths that are not covered here. I try to anticipate as many of the
questions as possible in writing these notes up, but the reality is that I
can’t anticipate all the questions.
Sometimes a very good question gets asked in class that leads to
insights that I’ve not included here.
You should always talk to someone who was in class on the day you
missed and compare these notes to their notes and see what the differences
are.

- This
is somewhat related to the previous three items, but is important enough
to merit its own item. THESE NOTES
ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!!
Using these notes as a substitute for class is liable to get you in
trouble. As already noted not everything in these notes is covered in
class and often material or insights not in these notes is covered in
class.

Here is a listing and brief description of the material in
this set of notes.

**Systems
of Equations and Matrices**

**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.

**Determinants**

**The Determinant Function** We will give the formal definition of the
determinant in this section. We’ll also
give formulas for computing determinants of and matrices.

**Properties of Determinants** Here we will take a look at quite a few
properties of the determinant function.
Included are formulas for determinants of triangular matrices.

**The Method of Cofactors** In this section we’ll take a look at the first
of two methods form computing determinants of general matrices.

**Using
Row Reduction to Find Determinants** Here we will take a look at the second method
for computing determinants in general.

**Cramer’s Rule** We will take a look at yet another method for
solving systems. This method will
involve the use of determinants.

**Euclidean
***n*-space

**Vectors** In this section we’ll introduce vectors in
2-space and 3-space as well as some of the important ideas about them.

**Dot Product & Cross Product** Here we’ll look at the dot product and the
cross product, two important products for vectors. We’ll also take a look at an application of
the dot product.

**Euclidean ***n*-Space We’ll introduce the idea of Euclidean *n*-space in this section and extend many
of the ideas of the previous two sections.

**Linear Transformations** In this section we’ll introduce the topic of
linear transformations and look at many of their properties.

**Examples of Linear Transformations** We’ll take a look at quite a few examples of
linear transformations in this section.

**Vector Spaces**

**Vector Spaces** In this section we’ll formally define vectors
and vector spaces.

**Subspaces** Here we will be looking at vector spaces that
live inside of other vector spaces.

**Span** The concept of the span of a set of vectors
will be investigated in this section.

**Linear Independence** Here we will take a look at what it means for
a set of vectors to be linearly independent or linearly dependent.

**Basis and Dimension** We’ll be looking at the idea of a set of basis
vectors and the dimension of a vector space.

**Change of Basis** In this section we will see how to change the
set of basis vectors for a vector space.

**Fundamental Subspaces** Here we will take a look at some of the
fundamental subspaces of a matrix, including the row space, column space and
null space.

**Inner Product Spaces** We will be looking at a special kind of vector
spaces in this section as well as define the inner product.

**Orthonormal Basis** In this section we will develop and use the
Gram-Schmidt process for constructing an orthogonal/orthonormal basis for an
inner product space.

**Least Squares** In this section we’ll take a look at an
application of some of the ideas that we will be discussing in this chapter.

*QR*-Decomposition
Here we will take a look at the *QR*-Decomposition for a matrix and how it
can be used in the least squares process.

**Orthogonal Matrices** We will take a look at a special kind of
matrix, the orthogonal matrix, in this section.

**Eigenvalues and Eigenvectors**

**Review of Determinants** In this section we’ll do a quick review of
determinants.

**Eigenvalues and Eigenvectors** Here we will take a look at the main section
in this chapter. We’ll be looking at the
concept of Eigenvalues and Eigenvectors.

**Diagonalization** We’ll be looking at diagonalizable matrices in
this section.