TITLE: Other guises for the vector space $\mathbb{R}^n$?
QUESTION [15 upvotes]: One way the vector space $\mathbb{R}^n$ can come up is as the space of polynomials over $\mathbb{R}$ of degree at most $(n-1)$ .  Here we have the isomorphism: $$(a_0,a_1,\ldots,a_{n-1}) \leftrightarrow a_0+a_1x+a_2x^2+\cdots+a_{n-1}x^{n-1}.$$  The linear independence of the polynomials in $\{1,x,x^2,\ldots,x^{n-1}\}$ is why it is legitimate to equate the coefficients of polynomials.
Question: What are other guises for the vector space $\mathbb{R}^n$?
I.e., what are some other interesting vector spaces that are isomorphic to $\mathbb{R}^n$?

REPLY [2 votes]: Given fixed $x_0, x_1, x_2, \dots, x_n$, polynomials of degree at most $n$ can be written using values $(p(x_0),p(x_1),\dots,p(x_n))$. The basis vectors $e_i$ are polynomials such that $e_i(x_j) = \delta_{ij}$, and converting from a tuple to a polynomial is Lagrange interpolation. If the values $x_i$ are roots of unity, a connection to FFT can be seen.