What's Linear Algebra?

We didn't use a textbook for this class but the notes are online at: https://slate.math.ubc.ca/slate/Slate/2009-2010/Winter_Term1...

I personally have both of Strang's Linear algebra books (Linear Algebra and its Applications, Introduction to Linear Algebra), there is also David C Lay's Introduction to Algebra.

My earlier course used Schaum's outline to Beginning Linear Algebra. I found that it doesn't give any reason for studying it, but if you like pure mathematics it is cheap and focuses on that aspect of linear algebra.

Linear algebra is fundamentally about solving equations and properties of systems of equations. In the course I'm taking we first showed that linear algebra has huge applications by doing a quick and shallow overview of Google PageRank. http://en.wikipedia.org/wiki/PageRank

Then we started from the beginning of linear algebra basically with a system of equations and finding their solutions. We used this idea to solve the problem of interpolation. Given a set of points how do we draw nice smooth points that connect them all, the solution that we looked at was a cubic spline. Before computers people would do this by hand using PHYSICAL 'calculators' in the form of a piece of wood/plastic that bends smoothly. http://pages.cs.wisc.edu/~deboor/draftspline.html

We then went on to cover the 'four fundamental subspaces' of a matrix. It is hard to describe but you can think of a matrix as encoding all of the information in a system of equations. The fundamental subspaces represent information about the different solutions that may or may not exist and where to find them in a 'space'. We used these ideas to look at resistor networks. One thing we modeled was the atmosphere as a very large graph all connected by resistors, we then simulated lightning as it 'traveled' through the matrix. From this we 'learned' that the atmosphere is most likely composed of elements with varying degrees of resistance since lightning is jagged shaped and not even. (Hand waving explanation there).

Then we came to the idea of orthogonality. If you remember a bit of about vectors two orthogonal vectors are vectors with a 90 degree angle between them. Using the ideas of orthogonality you can do least squares. A huge idea in statistics for linear regression, fitting a bunch of points to a straight line.

We also looked at the most awesome applied mathematical discovery the Fourier series/transform. The idea is that we can take a signal and transform it into a signal composed only of juxtaposed sin and cos waves. This is important in image analysis (i.e. medical imaging) as well as many other branches.

We then looked at eigenvalues and eigenvectors. When an matrix is applied (multiplied) to one of its corresponding eigenvector the eigenvector is only stretched or flipped. This is used in image manipulation, like properly stretching a picture without deforming it. It is the underpinning of Principle Component Analysis, a statistical technique to take data that has high-dimension (i.e. a lot of different measurements) to find the main directions where the measurements change.

I am personally doing research with a group using eigenvectors and eigenvalues to build algorithms that can be used to classify observations from a flow cytometer to differentiate between cells with or without cancer or hiv.

In my personal opinion linear algebra is just as important if not more so than calculus. Almost every aspect of modern analytical research needs the use of linear algebra as we are dealing with large amounts of information that must be manipulated with computers. This is when linear algebra is at its best (in my opinion).

Hope that answers some of your questions. Please do not take everything I've said as perfect or accurate, like I said I haven't done my final in the course yet, nor am I a mathematician. But I do hope that it shows that linear algebra has wide applications from basic cell research to how you get your daily dose of information.

An example: Control Systems. You have a device that takes some inputs and has some outputs. This device is usually modeled by a nxm matrix because it has n inputs and m outputs. From there, you want to build some other device to make sure that your original device does what you want it to do. Doing all this stuff requires linear algebra.

In general, linear algebra comes into play once the number of variables you're dealing with is greater than one. You mentioned that you've discovered that probability and statistics are important. Well, you'll need linear algebra to deal with that once the number of variables exceeds one.

Well, perhaps 'need' is too strong a word for it, but in the presence of multiple variables linear algebra is necessary as much as, say, integration for calculating the area underneath a curve. You can get by without it, but it has many advantages and brings more to the table.

Outside science/engineering domains? I can't think of anything it would be really useful for.