PlanetMath: finite-dimensional linear problem

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finite-dimensional linear problem

(Definition)

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

$\displaystyle L(u)=v,$

where $u\in U$ is the unknown, can be solved by means of row reduction. To do so, we need to choose a basis $a_1,\ldots, a_m$ of the domain

, and a basis $b_1,\ldots, b_n$ of the codomain

. Let

be the $n\times m$ transformation matrix of

relative to these bases, and let $y\in\mathbb{R}^n$ be the coordinate vector of

relative to the basis of

. Expressing this in terms of matrix notation, we have

$\displaystyle \begin{bmatrix}L(a_1),\ldots, L(a_m)\end{bmatrix} = \begin{bmatri... ... & M_{1m} \\ \vdots & \ddots & \vdots \\ M_{n1} & \ldots & M_{nm}\end{bmatrix},$
$\displaystyle v = \begin{bmatrix}b_1,\ldots, b_n\end{bmatrix} \begin{bmatrix}y_1\\ \vdots \\ y_n\end{bmatrix}$

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

$\displaystyle Mx =y,$

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

linear equations

$\begin{displaymath} \begin{array}{ccccl} M_{11} x_1 + &\cdots &+ M_{1m}\, x_m &=... ...ots\ M_{n1} x_1 + &\cdots& + M_{nm}\, x_m &=& y_n \end{array}\end{displaymath}$

with $x_1,\ldots, x_m$ unknown. Solutions $u\in U$ of the abstract linear equation

are in one-to-one correspondence with solutions of the matrix-vector equation

. The correspondence is given by

$\displaystyle u = \begin{bmatrix}a_1,\ldots, a_m\end{bmatrix} \begin{bmatrix}x_1 \\ \vdots \\ x_m\end{bmatrix}.$

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 focus 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 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.

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Also defines:

system of linear equations

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Cross-references: polynomial interpolation, point, polynomial, coefficient, rank-nullity theorem, rank, image, kernel, multiple, inconsistent, over-determined, under-determined, variables, dimension, one-to-one correspondence, solutions, equation, terms, vector, coordinate, bases, matrix, transformation, basis, row reduction, linear equation, finite-dimensional, codomain, domain, linear mapping
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This is version 9 of finite-dimensional linear problem, born on 2002-02-22, modified 2007-03-27.
Object id is 2502, canonical name is FiniteDimensionalLinearProblem.
Accessed 9122 times total.

Classification:

AMS MSC:

15A06 (Linear and multilinear algebra; matrix theory :: Linear equations)

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