# Algebraic properites of the direct sum

Bipolarity
Is the following statement true? I am trying to see if I can use it as a lemma for a larger proof:

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

Gold Member
Ask yourself whether the Wi are subspaces of W

Bipolarity
Ask yourself whether the Wi are subspaces of W

It is obvious if they are subspaces of ##W##, but what if they aren't?

BiP

Gold Member
It is obvious if they are subspaces of ##W##, but what if they aren't?

BiP

If W is the direct sum of the Wi then show me an element of one of the Wi's that is not in W

Is the following statement true? I am trying to see if I can use it as a lemma for a larger proof:

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
As it stands, the answer is obviously yes, since each ##W_i## is a subspace of ##W##, and hence ##W_ i\cap W=W_i##.

But I assume that there is a typo and that you meant ## W_{1} \bigoplus W_{2} \bigoplus ... \bigoplus W_{k} = V ##. Then, the answer is no. It is almost trivial to find a counterexample in R2.