It’s becoming a bit more difficult to write these, with grading and homework taking up much of my time, and I no longer am officially "taking" AGCA for credit.  Nonetheless, it’s an extremely interesting topic, and I’m going to go along with it for as long as I can.  I should have noted in the previous post: a ring R will always mean "commutative, with unit."

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Okay, first, I’m going to abbreviate Algebraic Geometry + Commutative Algebra as AGCA, since, you know, it’s a lot to say.  Second, these posts (with the title AGCA) may require a bit more mathematical maturity and abstract algebra than the others.

This is following the lectures of Professor Vitter, and, unfortunately, is only going to be a shallow copy with, most likely, more typos.  My intention is to learn the material by teaching, and to clear up and expand on the proofs given.  I do not pretend to be anywhere near as good a lecturer or mathematician, and this post is essentially “for entertainment purposes only.”  Reader beware.

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I’m sorry for the long delay in writing — I had a jordan-normal form post, but I made a few errors, and, thus, am saving it for when I have more time to go over Jordan form. 

I am now officially starting up grad school, and I will be attending classes in analysis, complex analysis, topology, and an algebraic geometry class.  I will be TA-ing a calculus class, and a probability and statistics class.  Because this mathblog (mlog?) is, for me, a tool of learning by teaching, I will begin to regurgitate things that I am learning in class, and will address problems that a significant number of my students will have.  This is not to say I will be solving any homework problems — ’cause I won’t!  But, as my old calculus teacher always said,

"If most of the class listens and studies but still doesn’t understand the lesson then it doesn’t make them bad students so much as it makes me a bad teacher."

Does this mean I’m abandoning differential topology?  Potentially; I need to see how much time I have left. 

Okay, so, we know (or you should know!) what cross and dot products are for Euclidean spaces.  We use them all the time!  On the daily!  So, I’m sure you’ve seen the equalities

a\cdot b = \|a\|\|b\|\cos(\theta)

\|a\times b\| = \|a\|\|b\|\sin(\theta)

where \theta is the angle between the vectors a and b.  Where do these equalities come from?  Let’s find out.

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I’ve been in the process of moving to New Orleans, so there hasn’t been much updating recently.  In fact, I haven’t been able to do much math at all!  I know, very sad.  I will be posting, next, about some random multivariable topics (primarily those that I need for my placement exam.  ha.) and also beginning to follow the (short, but concise) Topology from the Differentiable Viewpoint by Milnor. 

Lemme just note something quickly about TftDV: it’s an older book, but it comes highly recommended and, as far as I know, is still used relatively frequently as an introduction to the subject.  The problem here is that I learned the subject from a different (and slightly stricter) book; so whereas Milnor talks about manifolds sitting in {\mathbb R}^{m}, I will be talking about maps from the manifold M to {\mathbb R}^{m} explicitly and not assuming the manifold is sitting inside of real space.  

Also note that I am in no way an expert in this field.  Far from it, in fact.  I will be learning along with you!  If I make a careless error, please don’t hesitate to correct me.  In addition, I will most likely be going slightly slower and explaining in more detail many of the topics in the book and simply skipping over or mentioning briefly some of the other topics which I feel are slightly less important or slightly less interesting.  I will also (and this is why I haven’t posted it yet…) be experimenting with scanning pictures that I’ve drawn to illustrate proofs and things, and placing them on this blog.  I don’t have a scanner yet, and I feel that explaining open sets and also manifold mappings and things will be much, much clearer with a picture, so I’ve put it off.

I mean, okay, we wrote about the real spectral theorem, but how much do we really know about it?  A lot, I hope!  I hope you remember that we needed T to be self-adjoint in order to imply that we have an orthonormal basis of eigenvectors with respect to T!

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Current Update!

August 4, 2010

After the last post on the real spectral theorem, I don’t have too much more to do in linear algebra.  I think this is around all I need to study for my placement exams as far as linear algebra is concerned — and so, for now, this is where I’ll probably end.  I’ve yet to post an example of using the spectral theorems, and I may post something about using complex vector spaces (since I found a bit about how to define them, etc, etc.) but for the most part, I’m not gonna write too much more about linear algebra.

On the other hand, I haven’t really even begun to study multivariable, so I’ve begun by putting up some random tidbits of information.  The reason for this is twofold: because lengthy and well-laid-out texts for multivariable calculus are available (for example, I use Stewart‘s calculus now, but there are many others.) and, in addition, because these concepts as a whole take a ton of time and patience to develop.  I have neither.  Consequently, I will be posting about little things that I find interesting or, perhaps, tricky.  Anything that takes me a little while to "figure out" from Stewart’s text, I will post on here. 

What about my claim (a while ago!) that we’re going to use linear algebra to study groups?  This is still true.  I don’t have my representation theory book anymore, but it’s being sent to me eventually; when I have it, I can begin writing about it.

Last, all I want is to you guys to have fun!  So if you have a topic you want me to explore, tell me!  Please.  I like hearing from you guys!

This is going to be a short post about basic differentiating and integrating of multivariable functions.  Remember, just because something is polar doesn’t mean that it’s not fun! But, if something is fun, then it rarely is polar.  Just saying.

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Okay, so, last time we talked about the spectral theorem for complex vector spaces.  What did it say?  Do you remember?  Don’t look.  Fine, look.  Either way, it said that we have an orthonormal basis made of eigenvectors of some linear map T if and only if T is normal.  Now, being normal is not that big of’a deal.  It just means that TT^{\ast} = T^{\ast}T.  Not a biggie, right?  Yeah.

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