📐 Subspace Test

A non-empty subset $W \\subseteq V$ is a subspace if and only if for all $u, v \\in W$ and $c \\in F$: (1) $u + v \\in W$ and (2) $c \\cdot u \\in W$.

Proof: **(⇒)** If $W$ is a subspace, it is closed under addition and scalar multiplication by definition.

**(⇐)** Assume conditions (1) and (2) hold.

- **Zero vector:** Since $W$ is non-empty, there exists $w \\in W$. By (2), $0 \\cdot w = \\mathbf{0} \\in W$.

- **Additive inverse:** For $w \\in W$, by...

From: Advanced Linear Algebra

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