Best way to prepare for Linear Algebra?

MathWarrior

I just was wondering what would be the best way to begin preparing for a course in linear algebra. It is a upper division applied course. Also, how does linear algebra compare in, conceptual thinking compared to say Calculus 3?

The course description is as follows:
Solving linear systems, matrices, determinants, vector spaces, bases, linear transformations, eigenvectors, norms, inner products, decompositions, applications. Problem solving using MATLAB.

Also, does anyone have any recommendations of supplemental texts that they found especially helpful when taking linear algebra?

Choragos

Gilbert Strang's book, Linear Algebra and its Applications, is an excellent reference. You can find earlier editions online for dirt cheap (I think I paid 6 bucks for my copy). There's also another, more applied book called Matrix computations by Golub and Van Loan.

As far as preparation, I found linear algebra to be very different from other math disciplines. This is a dangerous statement since I use linear algebra every day, but I find that it's easier than most of the other math fields. Some of the mechanics can be tricky, but that's just turning the crank. Learning to think with general vector spaces is also a stretch, but once you get that idea so much of math becomes clearer. If the prof is good, you shouldn't have any problems.

I will give a caveat: I took linear algebra in undergrad and failed it badly because the teaching style and my learning style were way off. I wound up teaching myself from books much later as a grad student.

symbolipoint

Would you give more specific information about your preparation?

Would you explain more about the teacher's teaching style and your learning style? What did you do through your books when teaching yourself which were different from when you studied in class?

Choragos

After re-reading what I wrote, I was definitely unclear. I really didn't prepare at all. What I mean is, so long as your prof is reasonable, I don't think much preparation is necessary. However, pre-reading (or skimming) the books I suggested wouldn't hurt at all.

One thing I might suggest is while you're prepping for the class, have access to Matlab, Octave, Python, or something where you can run quick numerical examples. I find that helps a ton to verify a concept.


The prof in that case was on another plane. He was exceptionally brilliant, and every concept was trivial to him. His style was to state the concept and move on. I need to see examples and work through things myself. If I hadn't been taking 18 credits that semester, I probably would have had time to independently find and work through examples, and might have been fine. However, I didn't take the time to do that and had to withdraw.

Later, when I 'taught' myself, I had two strategies. One, I would read books with plenty of examples, and run through them both algebraically and generate numerical examples. At the same time, I was working on research that required heavy use of linear algebra, and so I could directly see the applications of what I was learning. Basically I dove right in and it was sink or swim! Now, linear algebra is so critical and integral to what I do that it's strange to me to consider a time where I didn't think in this way.

I'm not sure this is helpful. I do HIGHLY recommend the books though. Let me know if I can try to clarify any of this further.

leroyjenkens

I'm about to fail linear algebra simply because the tests word things differently than the book. I can answer every question in the book, and it's pretty easy, but when the wording differs, I don't really know what to do. The proofs are annoying too. Just get ready to memorize lots confusing terminology. And learn how to RREF asap. I can't believe I may fail such an easy course. I swear this crap is easy, yet somehow I fail the tests. Calc 2 was a *****, and I got a C in that. God I'll just be glad to get out of it. If I never hear the word "matrix" again, it'll be too soon.

And some advice I'll give for working out the long matrix problems where you do one thing, then another, then another, then another, then another. Make sure your arithmetic is correct the first time. Teachers say that if you make a mistake in the first step, they won't keep counting off on the subsequent steps based on that one error in the beginning. But in linear algebra, you can make a mistake in the first step, and you can get a result you can't even work with, so you wasted time, and you have to go back to the first step again to get the right numbers. So this crap about only marking off on the beginning part isn't realistic.

Saladsamurai

Khan Academy has a lovely set of videos on the subject. They are usually less than 15 minutes per video which is really nice since you know you can squeeze one in here and there. But if you really want to prepare, use must practice.

Link to khan: http://www.khanacademy.org/math/linear-algebra

A simple text that is very practical is one by Anton: http://www.amazon.com/Elementary-Lin.../dp/0471170550

No matter what text you choose, practice will be your best help. Do the in-chapter example problems out, filling in the steps that the author omits. Then do the end of chapter problems. The more the better. Even if you do not get to far into a text or the videos, at least when you start classes, it will not be the first time you are seeing it. This will always put you at an advantage.

Good Luck

Jorriss

Is there only one problem in the book? If not, I can't imagine how you can answer every question in the book, presumably a mountain of different problems, and yet the professors wording causes complete blocks.

What do you want to do? Because matrices absolutely do not go away in science.

NegativeDept

My standard advice for every math/physics/engineering class: go to libraries and bookstores. Flip through books on the subject. If you find one that, to you personally, seems to make more sense - buy it! This seems expensive, but paying <$200 to not fail a class is an OK deal. And for many math/physics/engineering majors, Jorriss has a point: being competent at matrix manipulation could be essential for your future work.

For linear algebra, it's very helpful to prepare by doing simple practice problems with the basic axioms of vector spaces and inner products. I was always mediocre at algebra, but good at visualizing 2D and 3D things. Thanks to my vector-space training, I can think of matrices as "things that rotate and/or stretch vectors." That means I can use my geometric intuition to help me do linear algebra. The secret is something I call The Very Big Theorem of Linear Algebra:

Every ##n \times m## matrix represents a linear transformation from ##R^m## to ##R^n##. Every linear transformation from ##R^m## to ##R^n## can be represented by an ##n \times m## matrix. (Here ##R^n## means the real vector space of dimension ##n##, using the Euclidean/Pythagorean definition of vector magnitude.)

The Very Big Theorem is also great for people who are good at algebra but bad at visualizing vectors and rotations. It underlies huge chunks of linear algebra. Orthogonal matrices preserve vector norm. Diagonal matrices just stretch vectors. Singular matrices multiply at least one vector by zero. Self-adjoint matrices would be diagonal if you rotated to a different coordinate system. Determinants tell you whether boxes in a vector space get bigger or smaller when you transform then, and whether the transformation turns right-handed coordinates into left-handed coordinates. And so on...

Woopydalan

some things in linear algebra are pretty tough for the first time learner...I had a lot of difficulties (still do) understanding vector spaces and subspaces. I still haven't grasped the fact that you can express vector spaces as matrices and power series and stuff other than straight vectors, its a very strange thing to me.

Good news is the beginning of linear algebra is pretty simple (row operations, cramer's rule, etc) and then it becomes a lot more challenging when you get to eigenvectors and solving homogeneous equations with a given matrix.

Just my experience of it though

Oh yeah and get the TI 36X pro so that you can double check yourself to make sure your determinants were calculated correctly (there is a lot of mundane calculation in LA)

bp_psy

I think linear algebra is pretty basic ,extremely useful and very cool. If you have regular high school algebra and some basic geometry you will do fine if you put the time in it. If the course is very proof based there might be a problem though. There are about 2/3 tiers of linear algebra at the undergrad level and you should really do them in order.