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A subspace is the direct sum of two others

Source: Iran PPCE 2012-Linear Algebra exam-P1

February 16, 2012
vectorlinear algebralinear algebra unsolved

Problem Statement

Suppose that W,W1W,W_1 and W2W_2 are subspaces of a vector space VV such that V=W1⊕W2V=W_1\oplus W_2. Under what conditions we have
W=(W∩W1)⊕(W∩W2)W=(W\cap W_1)\oplus(W\cap W_2)?
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