# finite-dimensional linear problem

Let $L:U\to V$ be a linear mapping, and let $v\in V$ be given. When both the domain $U$ and codomain $V$ are finite-dimensional, a linear equation

$$L(u)=v,$$ |

where $u\in U$ is the unknown, can be solved by means of row reduction. To do so, we need to choose a basis ${a}_{1},\mathrm{\dots},{a}_{m}$ of the domain $U$, and a basis ${b}_{1},\mathrm{\dots},{b}_{n}$ of the codomain $V$. Let $M$ be the $n\times m$ transformation matrix of $L$ relative to these bases, and let $y\in {\mathbb{R}}^{n}$ be the coordinate vector of $v$ relative to the basis of $V$. Expressing this in terms of matrix notation, we have

$$\left[\begin{array}{c}\hfill L({a}_{1}),\mathrm{\dots},L({a}_{m})\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {b}_{1},\mathrm{\dots},{b}_{n}\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill {M}_{11}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {M}_{1m}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {M}_{n1}\hfill & \hfill \mathrm{\dots}\hfill & \hfill {M}_{nm}\hfill \end{array}\right],$$ | ||

$$v=\left[\begin{array}{c}\hfill {b}_{1},\mathrm{\dots},{b}_{n}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {y}_{1}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {y}_{n}\hfill \end{array}\right]$$ |

We can now restate the abstract linear equation as the matrix-vector equation

$$Mx=y,$$ |

with $x\in {\mathbb{R}}^{m}$ unknown, or equivalently, as the following system of $n$ linear equations

$$\begin{array}{ccccc}\hfill {M}_{11}{x}_{1}+\hfill & \hfill \mathrm{\cdots}\hfill & \hfill +{M}_{1m}{x}_{m}\hfill & \hfill =\hfill & {y}_{1}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill & & \mathrm{\vdots}\hfill \\ \hfill {M}_{n1}{x}_{1}+\hfill & \hfill \mathrm{\cdots}\hfill & \hfill +{M}_{nm}{x}_{m}\hfill & \hfill =\hfill & {y}_{n}\hfill \end{array}$$ |

with ${x}_{1},\mathrm{\dots},{x}_{m}$ unknown. Solutions $u\in U$ of the abstract linear equation $L(u)=v$ are in one-to-one correspondence with solutions of the matrix-vector equation $Mx=y$. The correspondence is given by

$$u=\left[\begin{array}{c}\hfill {a}_{1},\mathrm{\dots},{a}_{m}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {x}_{m}\hfill \end{array}\right].$$ |

Note that the dimension of the domain is the number of variables,
while the dimension of the codomain is the number of equations. The
equation is called under-determined or over-determined depending on
whether the former is greater than the latter, or vice versa. In
general, over-determined systems are inconsistent, while
under-determined ones have multiple solutions. However, this is a
“rule of thumb” only, and exceptions are not hard to find. A full
understanding of consistency, and multiple solutions relies on the
notions of kernel, image, rank, and is described by the rank-nullity
theorem^{}.

## Remark.

Elementary applications exclusively on the
coefficient matrix and the right-hand vector, and neglect to mention
the underlying linear mapping. This is unfortunate, because the
concept of a linear equation is much more general than the traditional
notion of “variables and equations”, and relies in an essential way
on the idea of a linear mapping. See the
example (http://planetmath.org/UnderDeterminedPolynomialInterpolation) on
polynomial^{} as a case in point. Polynomial interpolation
is a linear problem, but one that is specified abstractly, rather than
in terms of variables and equations.

Title | finite-dimensional linear problem |
---|---|

Canonical name | FinitedimensionalLinearProblem |

Date of creation | 2013-03-22 12:26:05 |

Last modified on | 2013-03-22 12:26:05 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 12 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 15A06 |

Related topic | LinearProblem |

Related topic | RankNullityTheorem |

Defines | system of linear equations |