I am trying to solve this problem:
Let W_1, W_2, W_3 be subspaces of a vector space, V.
Prove that W_1 ∩ (W_2 + ( W_1 ∩ W_3)) = (W_1 ∩ W_2) + (W_1 ∩ W_3).
Can someone help me show this? I have tried using Dedekind's law, but not sure it that is the way to go.
The attempt at a solution
I tried with in my mind very trivial case...can somebody please show me a more detailed solution with more steps?
This is what I did...Since a subspace is a set, the laws of set operations apply. I assume (not sure if this is a valid assumption) that + here is the same as "union".
Now intersection is distributive over union,
i.e. a∩(b+c) = a∩b + a∩c
so in this case,
W_1 ∩ (W_2 + ( W_1 ∩ W_3))
= (W_1 ∩ W_2) + (W_1 ∩ (W_1 ∩ W_3))
In the second term, I use the properties that intersection is associative, and W_1 ∩ W_1 = W_1, and that term becomes W_1 ∩ W_3 which proves the required result.
Now can anyone answer if this always holds and please show me a more detailed solution with more steps that would make more sense? Thanks...