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Let $V$ be a vector space and $W \subset V$ be a direct summand of $V$. If $W' \subset W$ then under what conditions is $W'$ a direct summand of $V$?

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    Is $W'$ a subset or a sub _space_? Every subspace of a vector space is a direct summand: take a basis for $W'$ and extend it to a basis for $V$.2012-02-23
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    [Of course, if $W'$ is not a subspace then there is no hope. Look at the definitions.]2012-02-23

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