linear equation

Let $L:U\rightarrow V$ be a linear mapping, and $v\in V$ an element of the codomain. A linear equation is a relation of the form,

 $L(u)=v,$

where $u\in U$ is to be considered as the unknown. The solution set of a linear equation is the set of $u\in U$ 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 set. Otherwise, the equation is called consistent.

The general solution of a linear equation has the form

 $u=u_{p}+u_{h},\quad u_{p},u_{h}\in U,$

where

 $L(u_{p})=v$

is a particular solution and where

 $L(u_{h})=0$

is any solution of the corresponding homogeneous 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

 $y^{\prime\prime}+y=0.$
 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