Non surjectivity implying no solutions to a system of inhomogeneous equations

In page 66 of Linear Algebra Done Right, we are given a system of linear inhomogeneous system of equations: \begin{align} \sum_{k=1}^{n}A_{1,k}x_k &= c_1 \\ &\vdots \\ \sum_{k=1}^{n}A_{m,k}x_k &= c_m \end{align}

So that's equivalent to $$T(x_1,...,x_n) = (\sum_{k=1}^{n}A_{1,k}x_k,...,\sum_{k=1}^{n}A_{m,k}x_k) =(c_1,...,c_m)$$, where $$T: \mathbf{F}^n \to \mathbf{F}^m$$. Then, Axler asks whether there is some $$c_1,...,c_m$$ such that the system has no solutions. In the explanation, he says "thus we want to know if range($$T$$) $$\neq \mathbf{F}^m$$." I'm not sure why knowing this implies whether there is some choice of $$c_1,...,c_m$$ that makes the system have no solutions, and why that fact is relevant to showing it has no solutions.

Suppose $$\operatorname{range}(T) \ne \mathbf F^m$$. By definition of $$T$$ we hence have $$\operatorname{range}(T) \subsetneq \mathbf F^m$$.
That is, there exists some $$\mathbf c \in \mathbf F^m$$ such that $$T\mathbf x \ne \bf c$$ for all $$\mathbf x \in \mathbf F^n$$.
Do you see why this means that $$\mathbf c = (c_1, \dots, c_m)$$ makes the system have no solutions?