I've just completed a course in linear algebra. I'm a physics undergraduate and I don't plan on taking an abstract algebra course. That said, I've been reading a little bit about it.
As I understand it, a vector space over a field F is a set V along with two operations, scalar multiplication (*) and vector addition (+), that satisfy the following conditions:
- Associativity of vector addition ... $u + (v + w) = (u + v) + w$
- Commutativity of vector addition ... $u + v = v + u$
- Identity element of vector addition ... $u + 0 = u$
- Inverse element of vector addition ... $u + (-u) = 0$
- Identity element of scalar multiplication ...$ 1*u = u$
- Distributivity of scalar multiplication ... $a*(u + v) = a*u + a*v$
- Closure ... If $u, v$ are in $V$, $c*u + d*v$ is also in $V$.
A group is a set $G$ along with an operation $(*)$ satisfying the following:
- Closure ... If $g, h$ are in $G$, then $g * h$ is also in $G$.
- Associativity ... $(g * h) * j = g * (h * j)$
- Identity element ... $g * e = e * g = g$
- For each $g$ in $G$, there exists $h$ such that $g * h = e$.
I have a few of questions:
- Are my definitions of vector spaces and groups correct?
- What's the key difference between vector spaces and groups? They seem very similar to me.
- I was told by my linear algebra professor that it's only a convenience to think of vectors as arrows in R^3 with a direction and magnitude. He said that vectors are a much more abstract and general concept. What then are vectors, really?
- Are there any sets that are both a vector space and a group?
Thank you very much in advance!