Algebraic properites of the direct sum

  • Thread starter Bipolarity
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  • #1
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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
 

Answers and Replies

  • #2
lavinia
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Ask yourself whether the Wi are subspaces of W
 
  • #3
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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
 
  • #4
lavinia
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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
 
  • #5
Erland
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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.
 

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