Let $V$ and $W$ be two real vector spaces, $v \in V$ and $w \in W$. I'm getting some trouble in the following problem:
Let $u_1$ and $u_2$ be two elements in the tensor $V \otimes W$ such that $u_1 + u_2 = v \otimes w$. I'm asking myself if this implies that
- $u_1 = v_1 \otimes w$ and $u_2 = v_2 \otimes w$ with $v_1 + v_2 = v$, or
- $u_1 = v \otimes w_1$ and $u_2 = v \otimes w_2$ with $w_1 + w_2 = w$.
This seems reasonable to happen, but I'm not so used to work with tensor products. I'd like some reference to study this kind of situation if possible.