Let an inhomogeneous system of linear equations in triangular form
-
with
over a field
be given, where the diagonal elements are all not
.
Then the solutions
![{\displaystyle {}(x_{1},\ldots ,x_{m},x_{m+1},\ldots ,x_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad211d0d7bf941bf00c469a2940de94a54ff293b)
are in bijection with the tuples
.
The
![{\displaystyle {}n-m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed29419b7f0afb3c3b914a814b7c0a5550e4958)
entries
![{\displaystyle {}x_{m+1},\ldots ,x_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f038622ec045d2d293744a9679ae2d8c92c9565f)
can be chosen arbitrarily, and they determine a unique solution, and every solution is of this form.