Linear Algebra

[Luke Setzer]

I am preparing to take a one semester course on linear algebra online through my local community college. We will be using Linear Algebra and Its Applications, 3e by Lay. Here are the chapters:

Ch 1: Linear Equations in Linear Algebra
Ch 2: Matrix Algebra
Ch 3: Determinants
Ch 4: Vector Spaces
Ch 5: Eigenvalues and Eigenvectors
Ch 6: Orthogonality and Least Squares
Ch 7: Symmetric Matrices and Quadratic Forms

I am already familiar with the the material in the first four chapters because of my mechanical and industrial engineering backgrounds. The last three chapters look tricky. I have the textbook and the solutions manual as well as the TI-Nspire calculator and software.

The course homework and tests are all computer based multiple choice with the latter monitored via proctored learning lab on campus.

I welcome all words of wisdom from those who have taken a course like this.

[peterz]

Here are the chapters:

Ch 1: Linear Equations in Linear Algebra
Ch 2: Matrix Algebra
Ch 3: Determinants
Ch 4: Vector Spaces
Ch 5: Eigenvalues and Eigenvectors
Ch 6: Orthogonality and Least Squares
Ch 7: Symmetric Matrices and Quadratic Forms

The last three chapters are the useful ones. In particular, for you, Ch. 5 is important because of its direct relevance to the Heisenberg formulation of quantum mechanics. Orthogonality is important to that as well. Most matrices of interest to QM are symmetric.

Good luck! I've never taken such a course on-line, and would likely be afraid to do so because there is no chance for students to study together and no way to have personal Q&A time with the instructor -- especially if you should get in trouble. Don't make the mistake of assuming that because you have had exposure to Chaps 1-4 that you have mastered them in the context of the course!

Again, best of luck -- it's a fascinating field to study.

-pz

[jeconnett]

Luke Setzer said he is going to take a linear algebra course, the textbook (by David Lay) for which will include:

Ch 5: Eigenvalues and Eigenvectors
Ch 6: Orthogonality and Least Squares
Ch 7: Symmetric Matrices and Quadratic Forms

I agree with Pete on this, though for different reasons. These in my view are the really valuable parts of linear algebra.
That is, they are the parts that are valuable for statistics. In my current course in statistical computing I spend quite
a lot of time on eigenvectors and eigenvalues and symmetric matrices, orthogonality and least squares - not so much
on quadratic forms, though it is there in the background. The important thing is to get a strong intuitive understanding
of eigenvectors and eigenvalues and their relationship to covariance matrices and maximum likelihood estimation (which
you will not get in this course). Understanding eigenvectors (and eigenfunctions) is a key to understanding QM. But
again, don't settle for just formal algebraic statements. Go for the geometric intuition.

Coincidentally, I knew David Lay as a prof in grad school - this was decades before he wrote the textbook, about which
I know nothing - I would recommend the old text on finite-dimensional vector spaces by Paul Halmos. A friend of mine
in our department is writing a linear algebra text also which will take a very geometric approach.

John C.

[Luke Setzer]

Thanks, guys.

I never really knew much about linear algebra outside of its coverage in Calculus II and its use to solve systems of equations and vector operations in various engineering classes. We used it a great deal in my master's program to solve systems of inequalities for operations research. (The simplex method proved a very hard concept around which to wrap my brain!)

After completing my master's program, I downloaded Gilbert Strang's MIT lectures on the subject and watched all of them. I had no idea how powerful linear algebra truly is until I watched those. So I decided to learn more about it by taking a formal class.

If I had it to do again, I would have taken it back in 1985!

I have experienced other online math courses and done well in them so I am confident in the effectiveness of the delivery method. What is nice about MathXL is that the user can work problems and the system will lead the user Socratically through examples if needed. It really is a powerful teaching tool.

[Luke Setzer]

Given that I am also starting an online MBA program the same day this class starts, I have decided to audit this class for now rather than take it for a grade.

For what it is worth, my long range goal is to "bone up" on my mathematics and physics using online tools such as MathXL in conjunction with online undergraduate courses such as Linear Algebra. This will lay a firm foundation for entering an online graduate program in Applied Physics via the Columbia Video Network. Visit http://www.cvn.columbia.edu/deg/app_physics.php to learn more.

I do not have a firm timeline for entry but do know that I plan to earn it in stages. I can earn a Certificate of Professional Achievement in Applied Mathematics followed by one in Applied Physics and then use those credits toward a graduate program. These all have to be completed within a span of five years so I have to time things rightly.

[Oakthicket]

Sounds like you have some ambitious plans. That's great. It takes a talented and dedicated person to continue with such intense self-development.

[Luke Setzer]

I am in Chapter 4 of Lay and this has to be the densest chapter so far. It has many highly abstract concepts -- vector spaces, etc. I downloaded the Kindle version of Schaum's Outline of Linear Algebra but I still feel like I am trying to run through molasses.

Our own former member David McMahon wrote Linear Algebra Demystified but I confess reluctance to purchase it based on the Amazon reviews which, ironically, suggest it has errors comparable to those that McMahon said of Mills!

[jeconnett]

Luke Setzer said:

I am in Chapter 4 of Lay and this has to be the densest chapter so far. It has many highly abstract concepts -- vector spaces, etc. I downloaded the Kindle version of Schaum's Outline of Linear Algebra but I still feel like I am trying to run through molasses.

Our own former member David McMahon wrote Linear Algebra Demystified but I confess reluctance to purchase it based on the Amazon reviews which, ironically, suggest it has errors comparable to those that McMahon said of Mills!

You're right - I looked at the Amazon reviews, and some of them are quite awful - there must be
a lot of errors, even errors in the errata. That's too bad. I have found the 'Demystified' series to be
not very impressive in general. They should do better editing. As has often been said here, Mills could
benefit from this also.

David Lay was a prof when I was in grad school at U Maryland. He was considered an excellent clear
teacher. I have not seen his book, but vector spaces is not the most advanced or abstract topic - far
from it - the chapter on eigenvectors and eigenvalues is likely to be harder, though again not particularly
abstract. Actually I would not regard vector spaces as an abstract topic at all - it is easy in general
to find concrete finite- or infinite-dimensional examples. You want abstract, look at group theory, category
theory, K-theory, or just plain mathematical logic. The best way to 'demystify' a topic like linear algebra
is to see it presented by an excellent live teacher.

John C.

[Luke Setzer]

I went to see the tutor today but he left sick!

I will talk to the instructor tonight during our online chat room session.

[Luke Setzer]

I took the final examination last Saturday morning and scored 100% on it. Had I taken this course for a grade and not as an audit, I would have earned a solid A. I really, really enjoyed this material and look forward to applying this new knowledge in future courses.

[markie]

Clearly, this Hydrino Forum has sharpened your mind Luke.
Congratulations!
Mark