Theorem

An inhomogenous system of linear equations with more equations than variables has no solution for some choice of constant terms.

Proof

We use the same notation as defineed in 3.26.

Define ${latex.inlineT: F^{n} \rightarrow F^{m}}. The equation ${latex.inlineT(x_{1}, ..., x_{n}) = (c_{1}, ..., c_{m})} is the same as a system of equations that is non homogenous(with the c’s representing the values on the right side of the equations).

We can rephrase the question as a question of “what condition ensures that T is not surjective”, in other words not onto(one of the c vectors has no mapping).

Well, if we set n smaller than m, we get that T is not surjective per 3.24.