Lost In Space

You know, the more I study linear algebra, the more fascinated I get by it. Sure, it's a stupidly hard subject, but I find myself enjoying math in a way which I haven't done since the start of this education.

I've gone through quite a few types of space - P-space, R-space, M-space, et cetera; and I just had an epiphany reading the very last chapter of the course. Again, I love how this course book is built - you go "huh?" for the most part of it and then it ties everything up so neatly all of a sudden.

Apparently, the book ends on, there is only need for one space of every given dimension. Meaning, there is only need for one three-dimensional space, there is only one two-dimensional space, and so on and so forth. Essentially, there exists a "vanilla"-space, in which all kinds of mathemathical phenomena can be described.

So, for example, all equations of the form a+bx+cx^2=0 exist in Vanilla Three. So do all three-dimensional vectors, as do all two-dimensional planes. Every straight line can be described in Vanilla Three. In short, pretty much everything that can be described with three or fewer distinct variables exists in Vanilla Three, if I've read the course book right.

And, to make this even more powerful - each Vanilla space can be described in a Vanilla space of greater dimension. Vanilla Two is nested in Vanilla Three. Vanilla Four contains both of them. As far as I can tell, what I've just learned is a tool for describing any mathemathical object that I've ever heard of.

This is pretty awesome.