Other guises for the vector space $\mathbb{R}^n$?

For a (real, but it doesn't matter much; see below) polynomial $Q(x)$ of positive degree, consider the set of proper rational functions with denominator $Q(x)$, i.e., $\{ P(x)/Q(x) \mid \operatorname{deg} P < \operatorname{deg} Q\}$. This is a real vector space of dimension $\operatorname{deg} Q$.

So what? Well, this leads to an easy proof of the Partial Fractions Decomposition. Namely, you can see that the PFD is really claiming the existence of a certain nice basis for this space, consisting of certain rational functions with denominator a power of a single irreducible polynomial. It's easy to see that the given putative basis elements are linearly independent and that there are $\operatorname{deg} Q$ of them. So they must span the whole space!

This idea is written up in detail in a one-page note available here. I admit though that it could be written up in a more undergraduate-friendly way: first I do the general case of the PFD over any ground field, which requires a little field theory (not much, but no field theory is part of most first linear algebra courses), then at the end I briefly indicate how the field theory can be eliminated over $\mathbb{C}$ and $\mathbb{R}$. This really is brief because at the time it was, for some reason (does this even sound like me? important that the entire argument fit on a single page.

The same general technique can be used to prove other theorems. Another good use of it is in the calculus of finite differences, with applications to explicit computations of the power sums $\sum_{n=1}^n n^k$. If I remember correctly, it can also be used to prove things like Hermite's Polynomial Interpolation Theorem. The idea of going from linear independence to spanning by knowing the (finite!) dimension of the ambient space seems like it should be emphasized more strongly in courses: it is a simple showpiece of the power of linear algebra.

Added: I've been asked to say more about the things I mentioned in the last paragraph. The linear algebra of the discrete derivative comes up in Section 4 and 5 of this (unfinished) note on discrete calculus. As for Hermite Interpolation, see Theorem 12.11 of these notes. If you do, you'll notice that the proof is missing. It would be easy to prove it using linear algebra in the above way. But I am not assuming linear algbra in these notes...so I'm not actually sure at the moment what proof to give!

As pointed out in the comments, any finite-dimensional real vector space is isomorphic to some $\mathbb R^n$. It may be interesting and instructive to consider such spaces where we don't have a more or less canonical basis to start with (and hence don't have an "obvious" isomorphism to $\mathbb R^n$).

In this respect "polynomials of degree $<n$" is a bad example as this ranslates to "polynomials $a_0+a_1x+\ldots a_{n-1}x$ with $a_0,\ldots,a_{n-1}\in\mathbb R$" and and immediately suggests us the desired isomorphism.

If we instead consider the vector space of twice diffrentiable functions $\mathbb f\colon \mathbb R\to\mathbb R$ satisfying $f''(x)+f(x)=0$ for all $x$, there is again a "nice" basis, given by $\sin x $ and $\cos x$, but now it may look less obvious - why should one pick these functions specifically? And in which order?

Similarly, conside the space of sequences $(a_n)_{n\in\mathbb N}$ satisfying the recursion $a_{n+2}=a_{n+1}+a_n$ for all $n$. This is a twodimensional space as well, the Fibonacci sequence being the most famous member. But there is again no strikingly obvious choice of basis.

The discrete Fourier transform is another interesting example because it is directly tied to modern technology. A digital microphone turned on monitors the air continuously, producing every second a sample $$\mathbf{x}=(x_0, x_1, x_2, \ldots x_{2N})\in \mathbb{R}^{2N+1}, $$ with each entry $x_h$ corresponding to sound intensity at time $\frac{j}{N}$ (typically, $2N+1$ is approximately 44000). Therefore the space of sound samples is (isomorphic to) $\mathbb{R}^{2N+1}$.

The linear algebra setting is useful because it allows to change basis: namely, one can express $\mathbf{x}$ as a linear combination $$x_h= a_0 + \sum_{k=1}^Na_k \cos\left(\frac{2\pi k}{N}h\right) + \sum_{k=1}^Nb_k \sin\left(\frac{2\pi k}{N}h\right), $$ that is $$\mathbf{x}=a_0\mathbf{1}+\sum_{k=1}^Na_k \mathbf{c}_k + \sum_{k=1}^N b_k \mathbf{s}_k, $$ where $\mathbf{1}$ is the sound of constant intensity and $\mathbf{c}_k$ and $\mathbf{s}_k$ correspond to elementary harmonics. This operation is called discrete Fourier transform (DFT).

This representation of the original sample is much better suited to human ear. Suppressing the coefficients corresponding to higher frequencies one obtains a sound that differs imperceptibly from the original, but which occupies much less space when stored on a digital medium. This is the principle on which audio compression standards (such as .mp3) work.

I find it remarkable that all of this is an application of basic linear algebra.

The space of diagonal matrices.

As DonAntonio says in the comments, any real vector space $V$ of dimension $n$ is isomorphic to $\mathbb{R}^n$. Once you choose a basis $v_1,\dotsc,v_n$ in $V$, you can write any vector $v\in V$ uniquely as $v=\lambda_1v_1+\dotsc+\lambda_nv_n$, and so $v\mapsto(\lambda_1,\dotsc,\lambda_n)$ is an isomorphism $V\to\mathbb{R}^n$.

A subtle but important point is that the isomorphism depends on the choice of basis, so it is dangerous to say that $V$ is $\mathbb{R}^n$; you don't know how to match the elements up until you have a basis of $V$ (and a basis of $\mathbb{R}^n$, but we could take the standard one by convention). Indeed, it is even possible to define a basis of $V$ as an isomorphism $\mathbb{R}^n\stackrel{\sim}{\to}V$.

There are cases in which two objects are isomorphic via a unique preferred isomorphism (i.e. a natural isomorphism) that doesn't depend on any choices, and in that case it makes more sense to call them the same rather than just isomorphic.