finite-dimensional linear problem

Let L:UV be a linear mapping, and let vV be given. When both the domain U and codomain V are finite-dimensional, a linear equation


where uU is the unknown, can be solved by means of row reduction. To do so, we need to choose a basis a1,,am of the domain U, and a basis b1,,bn of the codomain V. Let M be the n×m transformation matrix of L relative to these bases, and let yn be the coordinate vector of v relative to the basis of V. Expressing this in terms of matrix notation, we have


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


with xm unknown, or equivalently, as the following system of n linear equations


with x1,,xm unknown. Solutions uU 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


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 theoremMathworldPlanetmath.


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 ( on polynomialMathworldPlanetmath 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