What am I allowed to assume in questions like this? 
Suppose $T \in \mathcal{L}(V, W)$, and $(w_1, \ldots, w_m)$ is a basis of $\operatorname{range}(T)$. Prove that there exists $(\varphi_1, \ldots, \varphi_m) \in \mathcal{L}(V, \mathbf{F})$ such that $$ T(v) = \varphi_1(v)w_1 + \cdots + \varphi_m(v)w_m $$ for every $v \in V$.

$V$ and $W$ are both vector spaces over a field $\mathbf{F}$.
I always find questions like this hard to understand, and I never know what exactly I am 'allowed' to do. I see two ways of interpreting it:

*

*We define $(\varphi_1, \ldots, \varphi_m) \in \mathcal{L}(V, \mathbf{F})$. These linear maps cannot change, and must be valid for every $v \in V$ such that $T(v) = \varphi_1(v)w_1 + \cdots + \varphi_m(v)w_m$.


*For every $v \in V$, we must be able to define some $(\varphi_1, \ldots, \varphi_m) \in \mathcal{L}(V, \mathbf{F})$ such that $ T(v) = \varphi_1(v)w_1 + \cdots + \varphi_m(v)w_m $ holds. Note that key difference here being we can redefine $(\varphi_1, \ldots, \varphi_m)$ for each $v \in V$.
Clearly there is only one correct interpretation. Which one is it? And if the other interpration was intended, how would the question have been worded differently?
 A: The first interpretation is the correct one.
One way you can tell this is the order of the quantifiers. The statement is:

There exist $\phi_1,\ldots,\phi_m \in \mathcal L(V,\mathbf F)$ such that for all $v \in V$ the equation $T(v) = \phi_1(v) w_1 + ... + \phi_m(v) w_m$ holds.

In particular, the quantifier on $v$ coming after the quantifiers on the $\phi$'s tells you that the $\phi$'s cannot depend on $v$.
And then, like any existence proof, what you actually have to do, as you say, is to define $\phi_1,\ldots,\phi_m \in \mathcal L(V,\mathbf F)$ and then use their definitions to verify that the required equation holds (for all $v \in V$).
A: Your first interpretation is the correct one. I agree that the question statement is somewhat unclear, and would be more clear if it were written as follows:

Prove that there exists $(\phi_1, \dots$, such that for all $v\in V$, $T(v) = \dots$.

If the other interpretation had been intended, the question would be worded as

Prove that for any $v\in V$, there exists $(\phi_1, \dots$

or perhaps as

Fix $v\in V$. Prove that there exists $(\phi_1,\dots$

