Equivalent definitions of vectors

Equivalent definitions of vectors.

In maths a vector is an object that obeys some axioms of a vector space. But in physics a vector can be thought as an object which is invariant under rotations of the system's coordinates.

Is there a way to prove this equivalence?

Answer

What's happened is that you've been victimized by a standard physicist abuse of a non-distinction (I myself was a victim of this once) between a vector space, and a vector space that has been equipped with a certain notion of how elements of that space should transform (which is an additional mathematical input).

Both physicists and mathematicians use the term "vector" to mean an element of some vector space. In classical mechanics, for example, the velocity of a particle moving in three dimensions is modeled as an element of $\mathbb R^3$. In quantum mechanics, the state of a system is modeled as an element of a Hilbert space, a special kind of complex vector space that can either be finite-, or infinite-dimensional. There are lots of other examples as well.

So where does this idea of transformation come in?

Well in physics, it's relevant ask what happens to certain vectors when we do some physical transformation on the system we are studying. If we take the single particle and its velocity vector as an example, we could ask ourselves what happens to that mathematical vector $\mathbf v$ when we physically rotate the axes with respect to which we've determined its components. If we go out and do this in the real world, we find that the components of the position of the vector $\mathbf x$ get rotated, \begin{align} \mathbf x \to R\mathbf x \end{align} and nothing happens to our time measurements, so our determination of the velocity vector, which is just the time-derivative of the position vector, also get's rotated; \begin{align} \mathbf v \to R\mathbf v \end{align} Notice that there was no such notion of transformation built into the original mathematical description of velocity vectors as elements of $\mathbb R^3$, it's an extra consideration we make as physicists. In other words, what we're doing here is saying:

"Let's consider acting on elements of the vector space $\mathbb R^3$ by rotations in a particular way that is physically meaningful.

Once you have determined such a physically meaningful notion of transformation, you then ask

"If I build some vector out of other vectors that are defined to transform in this way, then will the resulting vector transform in the same way?"

This is precisely what we did with the position and velocity. We defined the position to change in a particular (physically motivated and meaningful) way under rotations, and then we noticed that if position transforms like this, then velocity does too.

If you want to mathematically formalize the additional notion of transforming vectors in vector spaces, then you would appeal to a mathematical field called representation theory in which one studies different mathematical ways that one can act on elements of vector spaces by elements of (among other things) groups. In particular, one considers, in addition to a vector space $V$ mappings $\rho:G\to \mathrm{GL}(V)$ from a group (or algebra, etc.) to the set of invertible, linear operators on a vector space $V$. One then chooses to study certain of these representations according to the physical context, namely according to which representation gives a type of transformation that is physically meaningful.

In the example above with position an velocity, we are implicitly considering the representation $\rho:\mathrm{SO}(3)\to \mathrm{GL}(\mathbb R^3)$ defined by $\rho(R) = R$, where here $\mathrm{SO}(3)$ denotes the group of rotations of three-dimensional Euclidean space, and we are then saying

"Define position vectors to transform as $\mathbf x \to \rho(R) \mathbf x = R\mathbf x$ under rotations, and now notice that velocity can be shown to transform in the same way."