Linear Subspace

Let $K$ be a field, let $V$ be a vector space over $K$, and let $0_{V}$ be its zero vector.

A subset $W$ of $V$ is a linear subspace of $V$ if

1. $0_{V}\in W$;
2. $u+v\in W$ for all $u,v\in W$;
3. $\lambda v\in W$ for all $\lambda\in K$ and $v\in W$.