Definition 1. Let $(V, F, \cdot)$ be a vector space. $(W, F, \cdot)$ is a vector subspace of $V$ if it is a vector space and $W \subseteq V$.
Proposition 1 (Vector Space Test). Let $V$ be a vector space over $F$ and $W \subseteq V$. $W$ is a subspace if:
- \[0 \in W.\]
-
(Closure under addition) For all $x$, $y \in W$,
\[x + y \in W.\] -
(Closure under scalar multiplication) For all $x \in W$ and $\lambda \in F$,
\[\lambda x \in W.\]
Proposition 2. Let $V$ be a vector space. For any set of subspaces $\mathcal{W}$, $\bigcap \mathcal{W}$ is also a subspace.
Proposition 3. Let $V$ be a vector space and $W$, $W’$ be subspaces. If $W \cup W’$ is a subspace, then either $W \subseteq W’$ or $W’ \subseteq W$.