Geometric App's of Linear Algebra

[philosophking]

I'm currently a freshman in linear algebra, getting ready for the final, and all has been going well. The algebra's pretty much intuitive for me; I tend to enjoy the abstract theoretical stuff :). Anyway, to my question.

My professor has done little to none as far as applying the ideas of linear algebra to, say, geometry, which is something that I've heard is done in some other linear algebra classes. There was a post earlier that matt grime answered in which he gave some insight into the geometric applications of eigenvectors/values/spanning sets.

I was wondering if anyone had some sources of some books or websites I could refer to in order to gain a better knowledge of the geometric applications of linear algebra. Although it really won't help me with my class or anything, my interest in the field pushes me to ask.

So besides the obvious Wolfram, any ideas? If you just want to start a general thread within this forum about it, that would be cool too :).

Thanks again

 

[matt grime]

How far do you want to go? Geometry, by which we mean algebraic geometry is inextricably linked with linear algebra. A good interesting and accesible book is a CUP graduate tecxt in Algebriac groups whose exact title escapes me (it's on my shelf at work, I'll give it to you tomorrow) that is also fairly cheap.

Here's some idea: what is the determinant of a matrix? How about the top volume form on the exterior algebra?

 

[philosophking]

Hm? I don't think I know what you mean by "exterior algebra". As for how far I want to go, I'm just looking for some of the basic applications of linear algebra to geometry. I'll bank on my library at school having the book you're giving me. Thank you very much though, I look forward to learning more.

 

[Hurkyl]

Or for something a little more elementary, (real) vectors and (real) matrices are to multivariable calculus as real numbers are to ordinary calculus, so that could be a more immediate direction (though you may need to take an advanced calculus or real analysis class to see this introduced) which will eventually lead to differential geometry.

 

[philosophking]

Oh ok, well I'm taking analysis next semester. Now that I mention that, I have a question about that also. My school offers Analysis I and II (which I'm taking next year), and also Real Analysis and Complex Analysis... where's the distinction? Do I learn some of the latter in the former, and then go more in-depth later in the latter?

 

[matt grime]

Carter, Segal, Macdonald

Lectures on Lie Groups and Lie Algebras

Emphasis that you should read the part on Lie Algebras (as they are vector spaces) and linear algebraic groups which count as some of the more tractable and elegant "power user" topics.

 

[philosophking]

Very nice, thank you. May I ask where this would be taught in college (what class)?

 

[matt grime]

You're in the US by the sound of it? In that case this is a 500 or 600 level course, mainly because there are other things deemed more important (Fluid dynamics and special relativity, I ask you!).

 

[philosophking]

Ha... are most people here not in the US? At any rate, a 500 or 600 level course in math? Because the other courses you mention, fluid dynamics and special relativity, are physics courses. What could be the name of the course in which I would learn this?

 

[matt grime]

I don't know where the majority of people are from, but I'm in the UK (but know how the US system names things). Fluid dynamics and such are applied maths, which perhaps some places only do in physics. Simply put the emphasis in the US appears to be on teaching you 'calc' in various guises that have little to do with higher level maths. After 4 years if you're lucky you might get to do a course entitled intoduction to commutative algebra or something. If you go to grad school or take those course early then look out for things called algebraic geometry, group theory, lie algebras, to be honest there's no unifomr naming system through different years at the same university, nevre mind different universities or even different countries.

Why don't you look at the list of lecture courses available at your university and their syllabi? If you're unsure what they might mean ask, but they are 500 or higher would by my first guess.

 

[philosophking]

Ok thanks. Our school doesn't have a graduate program in mathematics. The closest I think I'm going to get to what you mentioned is our two-semester algebra course, which concentrates a lot on groups on the second semester. There is a senior math seminar course, which I am going to try to fit in my schedule (I'm a math/economics double major so I have another set of courses to worry about!), and since those are self-selected topics I could probably talk to the faculty about algebraic geometry. Thanks again for your help.