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Linear system/Superposition principle/Section

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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

Then

is a solution of the system

Proof



Let be a field, and let

be an inhomogeneous linear system over , and let

be the corresponding homogeneous linear system. If is a solution of the inhomogeneous system and if is a solution of the homogeneous system, then is a solution of the inhomogeneous system.

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.