linear equation
Let be a linear mapping, and an element ofthe codomain. A linear equation isa relation of the form,
where is to be considered as the unknown. Thesolution set of a linear equation is the set of that satisfy theabove constraint, or to be more precise, the pre-image . The equation iscalled inconsistent if no solutions exist, that is, if the pre-image isthe empty set. Otherwise, the equation is called consistent.
The general solution ofa linear equation has the form
where
is a particular solution and where
is any solution of the corresponding homogeneous problem, i.e. anelement of the kernel of .
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
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 |