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

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?

• Every finite-dimensional space over $\Bbb R$ is isomorphic to $\Bbb R^n$ for some $n$, so in that sense there are no further examples. For a concrete example, consider, e.g., the vector space of real polynomials of degree $\leq k$ for a fixed $k$. Commented Nov 4, 2021 at 6:41
• The point is that $\Bbb R^n$ provides all geometric intuition. Commented Nov 4, 2021 at 6:46
• Commented Nov 4, 2021 at 8:41

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).