Vector Subspace of Real Vector Space

Theorem
Let $\R^n$ be a real vector space.

Let $\mathbb W \subset\R^n$.

Then $\mathbb W$ is a linear subspace of $\R^n$ iff:


 * $(1): \quad \mathbf 0 \in \mathbb W$, where $\mathbf 0$ is the zero vector with $n$ entries


 * $(2): \quad \mathbb W$ is closed under vector addition


 * $(3): \quad \mathbb W$ is closed under scalar multiplication.