Distributivity of subspaces

Question

I need to either give a proof or find a counterexample to a statement:

$$L+(M∩N) = (L + M)∩(L + N)$$

Where $L$,$M$,$N$ are subspaces of a vector space $V$. I could do $LHS⊆RHS$ proof, but I'm stuck with backwards proof. I would be really grateful if someone could help me out.

The reverse statement isn't true. Consider M the xy plane in $R^3$, N the line spanned by $(1,1,1)$, and L the z axis. — symplectomorphic, Dec 02 '16 at 10:09

score 12 · Accepted Answer · edited Dec 02 '16 at 10:13

Take $M=\mathbb R\times\{0\}$ and $N=\{0\}\times \mathbb R$. Then set $L=\{(x,x)|x\in\mathbb R\} $

Then, $M\cap N=\{(0,0)\}$ (should be obvious why) meaning that the left side is equal to $L+\{0\}=L$.

But $L+M = L+N = \mathbb R^2$ (this should be easy to prove), meaning the right side is equal to $\mathbb R^2\cap \mathbb R^2 = \mathbb R^2$.

Since $L\neq \mathbb R^2$, clearly, the equality does not hold, and you can only conclude that $LHS\subseteq RHS$.

Distributivity of subspaces

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