Answer to Question #297054 in Linear Algebra for Ram

Question

<e> Show that if P and Q are vector Subspaces of a vector space V then P ∩ Q is also a vector subspace of V.

Answer 1

Define $x,y\in P\cap Q$ and also $\alpha \in \mathbb{F}$.

we have to prove that $P\cap Q\neq \phi$ $\alpha(x+y) \in P\cap Q$

$\text{since P and Q are subspaces of V THEN , P }≠\phi \text{ and } Q≠\phi\implies P\cap Q\neq \phi$

$\text{since P and Q are subspaces of V }\implies \alpha(x+y) \in P \text{ and } \alpha(x+y) \in Q$

Hence, $\alpha(x+y) \in P\cap Q$

Thus, we may conclude that, $P\cap Q$ is a subspace of $V$