Intersection of Tensor Products

Question

I know that two sets $A$ and $B$ are equal if and only if

1. $A \subseteq B$
2. $B \subseteq A$

But, how can I apply this to show that

$(U_{1}\otimes V) \cap (U_{2}\otimes V) = (U_{1} \cap U_{2}) \otimes V$, where $U_{1}, U_{2}$ are subspaces of vector space $U$ and $V$ is a vector space.

Answer 1

Consider a basis for $U_1 \cap U_2$, then $U_1 \setminus U_2$, and $U_2 \setminus U_1$.

Take a basis $e_1,\dots e_k$ as a basis for $U_1 \cap U_2$. Extend by $e_{k+1},\dots,e_n$ for $U_1$, and by $u_{k+1},\dots,u_m$ for $U_2$.

For the former, $U_1 \otimes V$ has basis $e_i \otimes v_i$ with $e_1, \dots ,e_n$ some basis for $U_1$ and $v_1, \dots, v_j$ some basis for $V$.

Likewise, $U_2 \otimes V$ has a basis $e_1, \dots u_m$, and the two intersect precisely at $e_1\otimes v_k, \dots e_k \otimes v_i$ as $i$ ranges through all of $V$. This is precisely the basis for the RHS.