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单词 LinearEquation
释义

linear equation


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

L(u)=v,

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

The general solution ofa linear equation has the form

u=up+uh,up,uhU,

where

L(up)=v

is a particular solution and where

L(uh)=0

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

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

y′′+y=0.
Titlelinear equation
Canonical nameLinearEquation
Date of creation2013-03-22 12:25:59
Last modified on2013-03-22 12:25:59
Ownerrmilson (146)
Last modified byrmilson (146)
Numerical id8
Authorrmilson (146)
Entry typeDefinition
Classificationmsc 15A06
Synonymlinear problem
Synonymlinear system
Related topicHomogeneousLinearProblem
Related topicFiniteDimensionalLinearProblem
Definesconsistent
Definesinconsistent
Definesparticular solution
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