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linear equation
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(Definition)
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Let
 be a linear mapping, and  an element of the codomain. A linear equation is a relation of the form,
where  is to be considered as the unknown. The solution set of a linear equation is the set of  that satisfy the above constraint, or to be more precise, the pre-image  . 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
where
is a particular solution and where
is any solution of the corresponding homogeneous problem, i.e. an element of the kernel of  .
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
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"linear equation" is owned by rmilson.
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(view preamble)
Cross-references: linear differential equations, scope, mapping, finite-dimensional linear problems, focus, linear algebra, kernel, homogeneous, general solution, empty set, equation, solution, relation, codomain, linear mapping
There are 40 references to this entry.
This is version 5 of linear equation, born on 2002-02-22, modified 2007-03-27.
Object id is 2496, canonical name is LinearProblem.
Accessed 16388 times total.
Classification:
| AMS MSC: | 15A06 (Linear and multilinear algebra; matrix theory :: Linear equations) |
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Pending Errata and Addenda
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