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# 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

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

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15A06*no label found*

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## Comments

## The "linear equation" article is very theoretical

This is one of the reasons why I’d like to see

related questionsandrelated problemsshowing up in the sidebar. Bruce Lee’s question has a much more concrete and applied flavour. But we probably need a way to “glue” the question in over here.Initially, recategorizing and moving things seemed like a good feature for moderators – but as @rspuzio and I were discussing, maybe everyone on the site should have basic “moderator” abilities, e.g. for making attachments between things.