Linear inhomogeneous system/Strictly triangular/Solution/Fact

Let an inhomogeneous system of linear equations in triangular form

${\begin{matrix}a_{11}x_{1}&+a_{12}x_{2}&\ldots &+a_{1m}x_{m}&\ldots &+a_{1n}x_{n}&=&c_{1}\\0&a_{22}x_{2}&\ldots &\ldots &\ldots &+a_{2n}x_{n}&=&c_{2}\\\vdots &\ddots &\ddots &\vdots &\vdots &\vdots &=&\vdots \\0&\ldots &0&a_{mm}x_{m}&\ldots &+a_{mn}x_{n}&=&c_{m}\\\end{matrix}}$

with ${}m\leq n$ over a field ${}K$ be given, where the diagonal elements are all not ${}0$.

Then the solutions ${}(x_{1},\ldots ,x_{m},x_{m+1},\ldots ,x_{n})$ are in bijection with the tuples

${}(x_{m+1},\ldots ,x_{n})\in K^{n-m}$.

The ${}n-m$ entries ${}x_{m+1},\ldots ,x_{n}$ can be chosen arbitrarily, and they determine a unique solution, and every solution is of this form.

Proof, Write another proof