Preface
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.
1.
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.
2.
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.
3.
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.
4.
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.

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*Sign up*Linear
Independence
In the previous section we saw several examples of writing a particular vector as a linear
combination of other vectors.
However, as we saw in
Example 1(b)
of that section there is
sometimes more than one linear combination of the same set of vectors can be used for a given
vector. We also saw in the previous section that some sets of vectors,
{ }
12
,,,
n
S
=
v
vv
K
, can
span a vector space.

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