Summary of MIT's Linear Algebra. Lecture 3: Matrix Multiplication and Inverses
catonmat.net
catonmat.net
"Here is a magic formula for multiplying matrices. Here is what you can do with it. Here are some examples." Without any clue WHY you would want to!
The approach I like is to discuss what a linear function is, show how you can represent linear functions by matrices (given two finite dimensional vector spaces and bases for those vector spaces, there is a 1 to 1 correspondence between linear functions and matrices), then show how function composition turns into matrix multiplication. The common argument against this approach is that it demands too much from the students. However I disagree. To me it motivates the subject, results in fewer magic formulas, and it makes it easier later for people to see how to apply it in practice.
For instance take the associative law of matrix multiplication. It is straightforward that:
F o (G o H) (x) = F((G o H)(x)) = F(G(H(x)))
(F o G) o H (x) = (F o G) (H(x)) = F(G(H(x)))
Now try to write out a direct proof that matrix multiplication is associative. It will be much longer, much harder, and at the end you'll have no clue how someone could have ever thought up this definition or noticed that THIS definition satisfied the associative law.
After reading Axler's book you should go and grab a copy of Strang's "Introduction to Linear Algebra" to view the applied side of linear algebra (this is the book mentioned in the article).
Strang's book is a terrible choice for first textbook of linear algebra (for the reasons btilly laid out), but it presents very imortant view on linear algebra, provided that you have mastered the foundations enough to compensate for the lack of rigour.
But for many, having a well-practiced ability to do operations on matrices and vectors first (even if it's just a bunch of magic formuas) allows them to learn the more basic facts via experimentation, because they don't get "stuck" right at the beginning. My experience is that there are more of these people than there are of us.
Basically, teaching is hard. Don't take the fact that you dislike a particular method for a proof that it's flawed.
By contrast my experience has been that if I sit down with someone who has been having trouble with basic linear algebra before long you can practically see the lightbulb turn on as it "clicks" and they see how things connect and why the formula is what it is. Even if they only do concrete operations from then on, it is a win that they can figure out how it goes from first principles.
Certainly when I had free reign over an introductory linear algebra class, I taught the linear function approach and nobody had any difficulty with it. See http://bentilly.blogspot.com/2009/09/teaching-linear-algebra... for a description of how that experience went.
In my math classes, if I found myself forgetting the formulas, I found it useful to stop and think about what the formula is trying to accomplish, and re-derive it from scratch. That works, but you can't do it if you don't understand the point of the formula in the first place.
Sorry.
Strang's lectures don’t just give you rote formulas. Most of his lectures he clearly shows you where things came from (i.e. how and why Matrix Elimination works, etc...). One of the reasons why he shows the different ways to multiply matrices in this lecture is because he uses it in subsequent lectures to show how null-spaces and column spaces work.
His lectures are not the be-all and end-all to linear algebra, but (IMHO) it gives you a great intuitive feel for linear algebra. Also remember that his particular course (18.06) is focussed on engineering students who tend to do less theoretic proofs than maths and natural sciences folk.
I personally like to get an intuitive understanding of how things work before delving into theory (maybe it is just me). I am currently (slowly) busy with a textbook called Linear Algebra (by Friedberg, Insel and Spence). This is a great textbook that shows you how matrices are defined from fields, etc… (i.e. theoretical). Yet it would be extremely difficult to understand without having an intuitive idea that Strang’s lectures give you.
These can be really useful when doing linalg by hand if you're e.g. trying to massage a system into a form that you can solve / code. It's useful to see that "oh, I don't have to multiply these out, I can just regroup them in this way because of the col or row picture."
Here's my motivation for LinAlg: implementing physical simulation / graphics papers on the computer. Strang's sytle is a very good fit for that.
I'm interested in a more abstract style for AI / knowledge rep issues, but it's not the first motivation.
Er ... I should add that in Ontario / Canada we did matrix multiplication in high school, so the university "first course" wasn't really a first course. Not sure about now, though.
My firm belief from having learned and taught the subject is that the linear function approach is superior. For instance when you are discussing concrete algorithms for physical simulations, it makes it easy to connect your understanding of the physical process you are trying to model with the equations you write down. That is because you understand how the matrix of numbers represents what you are trying to do.
By contrast having people write down long calculations and gain practice eventually leads to competency through familiarity, but in my opinion it takes longer before the math starts "connecting" to mental models of what is being written down.
And theirs is pretty equivalent to CMU's Matrix Algebra (which covers a bit more than MIT)
The lectures are free to download (which is awesome to 3rd world folks like me :) )
I guess I would call it organic learning communities. And as momentum grows, these learning communities will change the way we think about education.
For me, it's an amazing opportunity to have a structure course presented after completing university. As an undergraduate, I never got the mathematics education that I now want.
When I watch one of these courses I actually download the syllabus and go through a class as if I were a student. I find that doing this really helps me get through the materials and complete the course.
The thing I like about MIT OpenCourseWear (OCW) is how complete the courses are. You can generally find all the related materials on the OCW website. A number of schools (Berkeley, UCLA) will post videos but fail to include documents that were passed out in class. Nothing is more frustrating than seeing everyone pickup a handout but not being able to get your hands on it.