# linear algebra – 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.