Math 24
Linear Algebra

Last updated March 10, 2016 13:22:23 EST

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Course Objectives: Passing through the halls of Kemeny Hall, you will overhear professors and students discussing a myriad of topics. The comment made most frequently in an explanation — independent of subject — is It's just linear algebra. Whatever this subject is, one can be certain that linear algebra is as fundamental to higher mathematics as counting is to arithmetic. But what is it?

I could use the words like linear independence, span, and linear transformation, but they would not mean very much. I could say one of the things with which we will come to grips is the notion of dimension. We all seem to know that the world we live in is three-dimensional, but what does that mean?, and why in the world do we need to understand spaces with more than three dimensions? even infinite dimensional spaces? An excellent question.

In your studies you have certainly heard the terms linear approximation or linear models. One inference is that these approximations or models (while far from perfect) are at least amenable to study, the tools of course being linear algebra. There are two linear algebra courses at Dartmouth, Math 22 and 24. The former is concerned more with applications and less so with abstraction. The latter is concerned more with coming to grips with abstract notions like dimension, and learning to justify assertions with rigorous proofs. But linear algebra is far too powerful and beautiful a tool not to see at least some of its amazing applications. I mean a hammer is a wonderful tool, but until you have hit a few nails without bending them, it is not terribly useful.

So we will do our best to split our time. Most of it will be helping you get comfortable with abstract notions, but in (at least) x-hours we will work on learning to write proofs and some applications. Applications may include computer graphics, a 'mathy' one might include projecting a three-dimensional object on to a plane, perhaps rotating the object and reprojecting it; a non-mathy one would be some part of a video game where some object goes spinning off. High dimensional applications include rankings, e.g., how does Netflix really make suggestions for what you would like to watch? How does Google rank pages returned from a search request, and many, many more.

If nothing else, linear algebra is a beautiful and elegant subject, and one that can easily convince you of the power of mathematics.

T. R. Shemanske
Last updated March 10, 2016