Answer to Question #297054 in Linear Algebra for Ram

Question #297054

<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.


1
Expert's answer
2022-02-20T17:28:06-0500

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

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

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

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

Hence, α(x+y)PQ\alpha(x+y) \in P\cap Q

Thus, we may conclude that, PQP\cap Q is a subspace of VV

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