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.