Zero Subspace has Dimension 0

Last updated Nov 1, 2022

Let $V$ be a Vector Space. Then $\dim {\mathbf{0}} = 0$ for $\mathbf{0} \in V$.

Recall that The Empty Set is Linearly Independent and $$\text{span} \emptyset = \bigcap\limits{W \subset V : S \subset W, W \text{ is a subspace of V}} = {\mathbf{0}},$$since every Vector Subspace of $V$ is a superset of $\emptyset$ and ${\mathbf{0}}$ is a Vector Subspace of every Vector Subspace of $V$. Thus $\emptyset$ is a Vector Space Basis for ${\mathbf{0}}$ and $\dim {\mathbf{0}} = |\emptyset| = 0$. $\blacksquare$

• Subspace Span