Vector Spaces and Groups 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!
 A: *

*Yes. But you don't need to add closure in these definitions For groups, for example, notice that an operation is, first of all, a function $\cdot :G \times G \to G$. And that its codomain is $G$ itself.

*A vector space is a $4-$tuple $(\mathcal{V},{\Bbb K}, +, \cdot)$, where $$+: \mathcal{V}^2 \to \mathcal{V} \quad \text{and} \quad\cdot:\mathbb{K} \times \mathcal{V} \to \mathcal{V}$$ are the operations. The structure of a vector space is much richer than that of a group. A vector space has two operations and a underlying a field, while a group is only the set with one operation (satisfying conditions you well know). Given a vector space $(\mathcal{V},{\Bbb K}, +, \cdot)$, $(\mathcal{V},+)$ is an abelian group, always. Answering 4. along, given a field $\Bbb K$, $\Bbb K^n$ is both a vector field and an additive group, with respect to the operations of $\Bbb K$.

*Vectors are elements of a vector space. It is just a name. Examples of vector spaces are:


*

*Polynomials with degree less or equal to $n$, with real coefficients: $\mathcal{P}_n(\Bbb R)$. 

*All continuous functions from $[0,1]$ to $\Bbb R$: $\mathscr{C}^0([0,1],\Bbb R)$  

*$\Bbb R^n$ itself.

*Matrices with real coefficients: $\mathbb{M}_{n \times m}(\Bbb R)$.



and a lot more stuff. I used $\Bbb R$ for concreteness, in general you can take an arbitrary field (for polynomials, matrices, etc). So a vector can be an arrow, a function, a polynomial, a matrix...
