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.

Corollary
Criterion $(1)$ may be replaced by:


 * $(1'): \quad \mathbb W \ne \varnothing$

that is, that $\mathbb W$ is non-empty.

Proof of Corollary
Suppose $\mathbf 0 \in \mathbb W$.

Then $\mathbb W$ contains an element and is non-empty.

Suppose $\mathbb W$ contains an element $\mathbf x \in \R^n$.

Then, by criterion $(3)$:


 * $0 \mathbf x \in \mathbb W$

where $0$ is the zero scalar.

But $0 \mathbf x = \mathbf 0$, so $\mathbf 0 \in \mathbb W$.