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Suppose that $W,W_1$ and $W_2$ are subspaces of a vector space $V$ such that $V=W_1\oplus W_2$. Under what conditions we have $W=(W\cap W_1)\oplus(W\cap W_2)$ ?

this was the firs problem of our exam. does anyone know what is the answer to this problem?

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Could you clarify please? Are you asking under what conditions on $W$ do we have $W = (W \cap W_1) \oplus (W \cap W_2)$? – Shaun Ault Feb 17 '12 at 14:31
I wrote exactly the question's statement. I don't know what condition was in mind of our teacher. – Goodarz Mehr Feb 17 '12 at 14:42

Either one of the following conditions is sufficient:

  1. $W=U_1+ U_2$ for some subspaces $U_1\subseteq W_1$ and $U_2\subseteq W_2$
  2. $\dim W=\dim(W\cap W_1)+\dim(W\cap W_2)$ (if $\dim W$ is finite)
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