Example of a finite-dimensional vector space, with an example of finite-dimensional subspace and a counterexample [closed]
Choose $V= \mathbb{R^3} $ , then $V$ is a finite dimensional linear space over $\mathbb{R}$
Then, $V$ itself an example of a subspace of dimension $3$ of $V$.
Choose any $2$ - dimensional plane or line in $ \mathbb{R^3} $ that doesn't passes through the origin to produce an example of a set that is not a subspace.
There are plenty of others, just try to construct a subset of $ \mathbb{R^3} $ that is not closed under linear combination of vectors in it.