Linear Algebra Done Wrong [pdf]

3D game development involves a lot of very basic linear algebra (small matrix operations) and the occasional application of advanced linear algebra (conjugate gradients and similar solvers).

Yes. It's huge in engineering, for example: http://commons.bcit.ca/math/examples/civil/linear_algebra/in...

According to one of my linear algebra professors from back when, a lot of computational linear algebra was developed by the Soviets in the mid-20th century as a way to carry out optimized central planning. So I guess another application would be, socialism.

No idea if this is true or not but it's a good story. Also goes a long way towards explaining why so many linear algebra textbooks are translated from Russian.

Linear algebra matters since we always prefer linearity to any other forms because of its simplicity. If something isn't linear, we should transform it into a linear form for further examination. That's what calculus does. That's what analysis does. That's the way how we think.

Here's an example from the finance world. Let's say that you can't get a closed form solution for some option you are trying to price. One of the methods you may turn to is a finite difference method. You are going to have a series of equations to solve for the value at each grid point. A good understanding of linear algebra is essential if you want to code up your solution. You will be dealing with sparse matrices and tridiagonal matrices.

Not me. Well, I'm a post-grad and my Ph.D. is quite mathsy. That doesn't make me a mathematician (I still believe that the title should be earned and I definitely have not yet earned it), but at least I've done some mathematics.

Anyway, I always find it much easier to move from the concrete to the abstract. But once you've mastered an abstraction, it does seem to become "concrete" in your brain and then you can build on it. Thus, with a good background in basic linear algebra, you can move on to tensor analysis, but good luck if you want to jump into tensors straight away.

To me, one of the interesting things about how mathematical results have been discovered historically is that they have often been accidental (or intended for something else).

eg. eigenvalues were originally looked at for use with quadratic forms, but people later found that its interesting properties (orthogonality, symmetry, etc) were useful for lots of other things.

There is often also a large gap in time between when a mathematician playing around with a problem discovers an interesting result and when it is actually applied to something useful.

eg. Euler's law regarding complex exponentials was developed around 1740, but it wasn't until around 1807 when Fourier used it in harmonic analysis and 1897 when Steinmetz started applying it to electrical engineering.

But you wouldn't know that from reading a textbook, which is (understandably) arranged in such a way that omits historical context and why people bothered to study it in the first place. Most linear algebra books are classic examples of how to introduce abstract topics without any context.

My abstract algebra book from college introduced the axioms and used them right away. But new concepts were introduced independently and afterwards related to the axioms.