Is the following statement true? I am trying to see if I can use it as a lemma for a larger proof:(adsbygoogle = window.adsbygoogle || []).push({});

Let ##V## be a vector space and let ##W, W_{1},W_{2}...W_{k} ## be subspaces of ##V##.

Suppose that ## W_{1} \bigoplus W_{2} \bigoplus ... \bigoplus W_{k} = W ##

Then is it always the case that:

## (W_{1} \cap W) \bigoplus (W_{2} \cap W) \bigoplus ... \bigoplus (W_{k} \cap W) = W ##

In essence, this is asking whether there is a distributive law compatible with the set intersection operation and the direct sum operation. I am only asking for a determination of whether the statement is true for false. I will work out the proof/counterexample for myself.

Thanks!

BiP

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# Algebraic properites of the direct sum

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