Linear inhomogeneous system/Strictly triangular/Solution/Fact

Let an inhomogeneous system of linear equations in triangular form

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

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

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

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

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