linear equation

Let L:UV be a linear mapping, and vV an element of the codomain. A linear equation is a relationMathworldPlanetmath of the form,


where uU is to be considered as the unknown. The solution set of a linear equation is the set of uU that satisfy the above constraint, or to be more precise, the pre-image L-1(v). The equation is called inconsistent if no solutions exist, that is, if the pre-image is the empty setMathworldPlanetmath. Otherwise, the equation is called consistent.

The general solution of a linear equation has the form




is a particular solution and where


is any solution of the corresponding homogeneousPlanetmathPlanetmathPlanetmath problem, i.e. an element of the kernel of L.

Notes. Elementary treatments of linear algebra focus almost exclusively on finite-dimensional linear problems. They neglect to mention the underlying mapping, preferring to focus instead on “variables and equations.” However, the scope of the general concept is considerably wider, e.g. linear differential equations such as

Title linear equation
Canonical name LinearEquation
Date of creation 2013-03-22 12:25:59
Last modified on 2013-03-22 12:25:59
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 8
Author rmilson (146)
Entry type Definition
Classification msc 15A06
Synonym linear problem
Synonym linear system
Related topic HomogeneousLinearProblem
Related topic FiniteDimensionalLinearProblem
Defines consistent
Defines inconsistent
Defines particular solution