Vector Spaces and Groups

Question

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:

1. Associativity of vector addition ... $u + (v + w) = (u + v) + w$
2. Commutativity of vector addition ... $u + v = v + u$
3. Identity element of vector addition ... $u + 0 = u$
4. Inverse element of vector addition ... $u + (-u) = 0$
5. Identity element of scalar multiplication ...$ 1*u = u$
6. Distributivity of scalar multiplication ... $a*(u + v) = a*u + a*v$
7. 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:

1. Closure ... If $g, h$ are in $G$, then $g * h$ is also in $G$.
2. Associativity ... $(g * h) * j = g * (h * j)$
3. Identity element ... $g * e = e * g = g$
4. For each $g$ in $G$, there exists $h$ such that $g * h = e$.

I have a few of questions:

1. Are my definitions of vector spaces and groups correct?
2. What's the key difference between vector spaces and groups? They seem very similar to me.
3. 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?
4. Are there any sets that are both a vector space and a group?

Thank you very much in advance!

Answer 1

1. 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.

2. 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 $\Bbb K$.

3. 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...