Intersection of Vector Subspaces is a Vector Subspace

Last updated Nov 1, 2022

Let $V$ be a Vector Space over Field $F$ and let ${W_{\alpha} \subset V}{\alpha \in A}$ be a collection of Vector Subspaces of $V$. Then $\bigcap\limits{\alpha \in A} W_{\alpha}$ is a Vector Subspace of $V$.

Let $\mathbf{a}, \mathbf{b} \in \bigcap\limits_{\alpha \in A} W_{\alpha}$ and let $c \in F$. Then $c \mathbf{a} + \mathbf{b} \in W_{\alpha}$ $\forall \alpha \in A$ since A Subset of a Vector Space is a Subspace iff it is closed under scaling and addition, so we have that $c \mathbf{a} + \mathbf{b} \in \bigcap\limits_{\alpha \in A} W_\alpha$. Because A Subset of a Vector Space is a Subspace iff it is closed under scaling and addition, we see $\bigcap\limits_{\alpha \in A} W_\alpha$ is a Vector Subspace of $V$. $\blacksquare$

1. Intersection of Structures is still a Structure.