Zero Subspace is Subspace

Theorem

Let $V$ be a vector space over $K$ with zero vector $\mathbf 0$.

The zero subspace $\set {\mathbf 0}$ is a subspace of $V$.

Proof

We use the Two-Step Vector Subspace Test.

$\set {\mathbf 0}$ is not empty, because it contains $\mathbf 0$.

$\set {\mathbf 0}$ is closed under $+$ because:

$\forall \mathbf x, \mathbf y \in \set {\mathbf 0}, \mathbf x + \mathbf y = \mathbf 0 + \mathbf 0 = \mathbf 0 \in \set {\mathbf 0}$

$\set {\mathbf 0}$ is closed under multiplication because:

$\forall \lambda \in K, \mathbf x \in \set {\mathbf 0}: \lambda \mathbf x = \lambda \mathbf 0 = \mathbf 0 \in \set {\mathbf 0}$

Hence the result, from the Two-Step Vector Subspace Test.

$\blacksquare$

Sources

• For a video presentation of the contents of this page, visit the Khan Academy.