Exercise.
Suppose $T \in \mathcal{L}(V,W)$ and $w_1,\ldots,w_m$ is a basis of $\text{range} \ T$. Hence for each $v \in V$, there exist unique numbers $\varphi_1(v), \ldots, \varphi_m(v)$ such that
$$ Tv = \varphi_1(v)w_1 + \cdots + \varphi_m(v)w_m, $$
thus defining functions $\varphi_1, \ldots, \varphi_m$ from $V$ to $\mathbb{F}$. Show that each of the functions $\varphi_1, \ldots, \varphi_m$ is a linear functional on $V$.
Source.
Linear Algebra Done Right, Sheldon Axler, 4th Edition, Exercise 5. in Section 3F.
My Question.
I was able to show they're linear functionals by using the linearity of $T$. But my question has to do with the assumption taken in the exercise, that there does indeed exist unique numbers $\varphi_1(v), \ldots, \varphi_m(v)$ such that $Tv = \varphi_1(v)w_1 + \cdots + \varphi_m(v)w_m$.
I understand why they're unique is because $w_1, \ldots, w_m$ is a basis, but how do we know they are functions of $v$? Would that be easy to show? I might be forgetting an old result in the text.
If you agree that for every $v\in V$, there are unique scalars $a_1$, $a_2$, $\dots$, $a_m$ such that $$Tv=a_1w_1+a_2w_2+\dots+a_mw_m$$ then we can define for each $1\leq i\le m$, a function $\phi_i\colon V\to \mathbb F$ as follows.
Remember what a function does is assigns a unique value in the codomain to each point in the domain. Take $v\in V$, then find unique scalars $a_i$s such that the above equation holds and then define $\phi_i(v)=a_i\in\mathbb F$.
Hence, the functions $\phi_i$ assigns a unique value in $\mathbb F$ to each point in $V$, and by definition of these functions, for any $v$ in $V$, we can write $$Tv=\phi_1(v)w_1+\phi_2(v)w_2+\dots+\phi_m(v)w_m$$
Hope this helps. :)