[Math] Vector Spaces and Groups

abstract-algebragroup-theorylinear algebravector-spaces

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!

Best Answer

  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...

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