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 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 convenient to think of vectors as arrows in $\Bbb R^3$ with a direction and magnitude. He said that vectors are a much more abstract and general concept. What, then, are vectors, really?
  4. Are any sets both a vector space and a group?

Thank you very much in advance!

  • 3
    $\begingroup$ A vector space is an abelian group with an action by a field. So every vector space is also a group. But not every group can be a vector space, for example if it is not abelian. $\endgroup$
    – ziggurism
    Commented Dec 14, 2014 at 4:32
  • $\begingroup$ You're going to need, at least, some reasonably advanced linear algebra and some Lie Groups theory in physics later, so you deciding not to take abstract algebra could be a bad idea. In my university, it was compulsory for physics student to take at least a basic course in group and ring theory. $\endgroup$
    – Timbuc
    Commented Dec 14, 2014 at 4:37
  • $\begingroup$ (3) Is the hard question, because there are lots of uses of vector spaces in quite a few cases. It's a generalization which simply crops up enough in mathematics to call them the same thing, and study them, but it is hard to give a visual sense of what is going on, since some fields don't have a visual intuition. $\endgroup$ Commented Dec 14, 2014 at 4:39
  • $\begingroup$ @Timbuc Thanks for the advice. At my university (University of Illinois at Urbana-Champaign) it's not compulsory for physics students to take abstract algebra. It is, however, compulsory for us to take linear algebra and complex analysis. The rest is up to us what we take. $\endgroup$
    – Cody
    Commented Dec 14, 2014 at 4:40
  • 2
    $\begingroup$ Timbuc's advice is not about compulsory, it is about what you want to do if you want to study physics. :) $\endgroup$ Commented Dec 14, 2014 at 4:40

1 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...


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