# Linar algebra: dimension of intersection

I suspect that $$dim((U_1+U_2)\cap U_3)=dim(U_1\cap U_3)+dim(U_2\cap U_3)$$ where $$U_1$$, $$U_2$$ and $$U_3$$ are vector spaces, but I couldn't find a proof or a counterexample. How could I proceed?

• What if $U_1=U_2=U_3$? Can the two sides be still equal? Jun 8 '19 at 19:33
• I didn't think the simplest counterexample, thanks Jun 8 '19 at 19:35