Tonight in linear algebra, we started talking about vector spaces. A vector space is a set of objects that you call "vectors" (whether they are traditional vectors or not does not matter), with "vector addition" and scalar "multiplication" defined however you want, and satisfying some axioms (like commutativity of the addition and having an additive identity and scalar multiplication distributing over the addition and so on).

According to our prof, this is the topic that differentiates* this course from the lower-level matrix algebra class that my school also offers.

Now, this is exactly the kind of thing I think is the most fun in math. "Here is this tiny set of rules - go see what it does." Those of you who read my Laguerre Planes paper of last semester may recall that I spent a lot of that paper discussing how various systems were, in fact, Laguerre Planes. I'm into that.

But, man, were people confused. Some of them had never had abstract algebra, or apparently even the rudiments thereof (I haven't taken the class myself), and so the idea of taking some arbitrary operation and calling it "addition" really screwed with their heads. Calling something that wasn't a vector a vector, and calling (-1, -1) "the zero vector" (as happened in one example) - all confusing.

A few semester ago - perhaps as long as two years ago - Ed told me that when you get to a certain point in math, it all starts to kind of become One Thing, and I feel like I'm reaching that point. I think what I feel is actually an illusion, but it seems like the same concepts and ideas come up over and over again in my classes. I can draw on multiple sources of knowledge for the same concepts.

When people in my classes don't "get" something that seems readily comprehensible to me, it is very hard for me not to think it's because they're stupid. Yet I know that comprehending a math concept is not completely, and probably not even mostly, about some generic intelligence. One way I can prove this to myself is to consider that I can remember being frustrated and confused by math at various points in my life, for instance many times in middle school, that I easily and completely understand now. And I have not actually gotten intrinsically smarter over time.

It feels to me like I've simply been exposed to a lot more math than some of my classmates. The kind of generic abstract algebra stuff that allows me to easily understand vector spaces comes to me from

1. Proofs (a prereq for linear algebra), where we did some abstract algebra

2. Discrete Math, where I think the idea of groups/fields came up somewhere

3. The first time I took linear algebra, where we had this exact topic, and

4. Higher Geometry II, where we did a lot of field/group stuff in the beginning

That's really quite a lot of times that I've been exposed to material that was similar enough to make this a cinch. It's no wonder I don't have any trouble with it.

(* oh no! calculus!)

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## 2 comments:

Perhaps you have found a way to major in logic after all...

Heh, yes. My classmates do better as we move from abstract to concrete (in calculus), and worse as we move from concret to abstract (in linear algebra), and I find I tend to be exactly the other way - better at the abstract stuff. Of course, that "better" may be relative to classmates in every case (like you were better at backing up around cones in driver's ed), but that's what counts, at least for now.

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