# linear equation

Let $L:U\to V$ 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

$$u={u}_{p}+{u}_{h},{u}_{p},{u}_{h}\in U,$$ |

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 |