I have the following problem in linear algebra:
Let $A$ be a vector space and $V$, $W$ subspaces of $A$ such that $V \cap W = \{0\}$:
Prove that $\operatorname{span}\{V, W\} := \{λ_1v + λ_2w : v \in V, w \in W, λ_1, λ_2 \in F\}$ is isomorphic to $V \oplus W$:
[Hint: Show that the function $T(v,w) = v + w$ is a linear isomorphism.]
My problem, oddly enough, is with the hint itself. I'm not sure how $T(v,w) = v + w$ can be an invertible linear operator. Without knowing either $v$ or $w$, there should be no way of retrieving them from the result of the operator. I also don't see how it can be used to map between $\operatorname{span}\{V,W\}$ and $V\oplus W$ (or vice versa). In effect, I don't really know where to get started with the problem.