I’m currently in a linear algebra class. As such, I will start with some definitions and go on from there. I provide links for things I do not have room to define. Note that the numbering corresponds to the actual numbering my professor uses during lectures. If some points are missing, it’s because I feel like they aren’t necessary. Let’s begin.
Definition 19.1
A vector space V over a field F is a set V equipped with compositions
+: V x V -> V, (x, y) |-> x+y (vector addition)
*: F x V -> V, (a, x) |-> a*x (scalar multiplication)
such that the following are satisfied for all a,b in F, x,y in V :
1. (V,+) is an abelian group.
2. a*(b*x) = (ab)*x (mixed associative law)
3. a*(x+y) = a*x + a*y and (a+b)*x = a*x + b*x
4. If 1 is the identity element of the field F, then 1 * x = x.
Note that since (V,+) is a group, it has its own identity element, the “zero vector”. This is not to be confused with the additive identity element “0” in the field F.
If F = R or F = C, then V is called a real or complex vector space, respectively.
Note that F^n is a vector space over F with the defined operations + and *. In particular, every field F is a vector space over itself.
I’ve wanted to start a math blog for a while. Mainly this is so I have a quick reference to theorems and proofs on the web so I don’t have to shuffle through my notes for every homework assignment. Of course, I’ll try to make the material not terribly dull. I’ll try to include my own comments on techniques used in a proof, etc.
Occasionally I’ll come across some piece of mathematics unrelated to what I’m currently studying in the classroom and post it here with comments.
Note that I am merely a first year undergraduate, so this is going to be far from Terence Tao’s blog.
