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, alinear equation

$$L(u)=v,$$

where $u\in U$ is the unknown,can be solved by means of row reduction. To do so, we needto 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$y\in {ℝ}^n$ be the coordinate vector of $v$ relative to thebasis of $V$. Expressing this in terms of matrix notation, we have

$$v=\left[\begin{array}{c}\hfill b_1,\mathrm{\dots },b_n\hfill \end{array}\right]\left[\begin{array}{c}\hfill y_1\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill y_n\hfill \end{array}\right]$$

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

$$Mx=y,$$

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

with $x_1,\mathrm{\dots },x_m$ unknown. Solutions $u\in U$ of the abstract linearequation $L(u)=v$ are in one-to-one correspondence with solutions ofthe matrix-vector equation $Mx=y$. The correspondence is given by

$$u=\left[\begin{array}{c}\hfill a_1,\mathrm{\dots },a_m\hfill \end{array}\right]\left[\begin{array}{c}\hfill x_1\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill x_m\hfill \end{array}\right].$$

Note that the dimension of the domain is the number of variables,while the dimension of the codomain is the number of equations. Theequation is called under-determined or over-determined depending onwhether the former is greater than the latter, or vice versa. Ingeneral, over-determined systems are inconsistent, whileunder-determined ones have multiple solutions. However, this is a“rule of thumb” only, and exceptions are not hard to find. A fullunderstanding of consistency, and multiple solutions relies on thenotions of kernel, image, rank, and is described by the rank-nullitytheorem.

Remark.

Elementary applications exclusively on thecoefficient matrix and the right-hand vector, and neglect to mentionthe underlying linear mapping. This is unfortunate, because theconcept of a linear equation is much more general than the traditionalnotion of “variables and equations”, and relies in an essential wayon the idea of a linear mapping. See theexample (http://planetmath.org/UnderDeterminedPolynomialInterpolation) onpolynomial as a case in point. Polynomial interpolationis a linear problem, but one that is specified abstractly, rather thanin terms of variables and equations.