Non-trivial examples of finite-dimensional vector spaces over $\mathbb{R}$

Question

I'm a first year undergraduate pursuing a degree in mathematics. So far in my linear algebra course, we have covered the abstract definition of a vector space over a field and linear maps, among other things.

I would like to know if there are any interesting examples of finite dimensional vector spaces over $\mathbb{R}$ aside from $\mathbb{R}^n$ for some $n \in \mathbb{N}$. Also, how would one visualise them in order to develop a geometric intuition for these non-trivial vector spaces?

Answer 1

Any finite dimensional vector space $V$ of dimension $n$ over $\mathbb{R}$ is isomorphic to $\mathbb{R^n}$ . ( $\exists$ a Invertible linear transformation)

I would like to know if there are any interesting examples of finite dimensional vector spaces over R aside from $\mathbb{ R^n}$ for some $n\in{\mathbb{N}}$

$\mathscr{P_n} =\{ a_0 +a_1 x+a_2 x^2+...+a_n x^n : a_i \in {\mathbb{R}},0\le i \le n\}$

Define, $+$ on $\mathscr{P_n}$ by

$\sum_{i=0}^{n}{a_i x_i}+ \sum_{i=0}^{n}{b_i x_i}= \sum_{i=0}^{n}{(a_i+b_i)x_i} $

Define, scalar multiplication by

$c(\sum_{i=0}^{n}{a_i x_i})=\sum_{i=0}^{n}{(ca_i) x_i}$

Then, $V=(\mathscr{P_n}, +, . )$ is a vector space over $\mathbb{R}$.(Check)

Here, $V$ is a finite dimensional vector space of dimension $n+1.$ (Check).