Vector Subspace of Real Vector Space

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Theorem

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

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

Then $\mathbb W$ is a linear subspace of $\R^n$ if and only if:

$(1): \quad \mathbf 0 \in \mathbb W$, 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:

$(1'): \quad \mathbb W \ne \O$

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

Proof

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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 $0 \mathbf x = \mathbf 0$ from Vector Scaled by Zero is Zero Vector, so $\mathbf 0 \in \mathbb W$.

$\blacksquare$

Also see

• Two-Step Vector Subspace Test
• Null Space is Subspace

Sources

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