I'm trying to understand a proof of the following statement:
Let $E_1, E_2$ be two subspaces of a vector space $E$. Then if $F$ is any other vector space, $(E_1\otimes F) \cap (E_2 \otimes F) = (E_1\cap E_2)\otimes F.$
The book that i'm using starts the proof by saying:
"Clearly, $(E_1\cap E_2)\otimes F \subset (E_1\otimes F) \cap (E_2 \otimes F) $ "...
and well, for me it's not that much clear. I've tried to justify it by myself, but i think that i'm going in the wrong direction, since it is not that much simple. What i've tried so far was to consider a basis to $E_1\cap E_2$ and to extend it to a basis for $E_1$ and $E_2$, respectively. And then writing $E_1\cap E_2 \otimes F$ as a direct sum of one-dimensional vector spaces i've tried to use some of the tensor properties to manage to show that any element of this vector space can be written by elements of the other one, but i was unsucessful.
Can someone give a good explanation? Thanks in advance. The other inclusion i could understand pretty well!