Linear system/Superposition principle/Section
Appearance
Let denote a matrix over a field . Let and denote two -tuples, and let be a solution of the linear system
and a solution of the system
is a solution of the system
Proof
See exercise.
Let be a field, and let
be an inhomogeneous linear system over , and let
This follows immediately from fact.
In particular, this means that when is the solution space of a homogeneous linear system, and when is one
(particular)
solution of an inhomogeneous linear system, then the mapping
gives a bijection between and the solution set of the inhomogeneous linear system.