finite-dimensional linear problem (original) (raw)

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

where u∈U 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 basisb1,…,bn of the codomain V. Let M be the n×mtransformation matrix of L relative to these bases, and lety∈ℝn 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 x∈ℝm unknown, or equivalently, as the following system of n linear equations

with x1,…,xm unknown. Solutions u∈U of the abstract linear equation L⁢(u)=v are in one-to-one correspondence with solutions of the matrix-vector equation M⁢x=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 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 theconcept 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) onpolynomial 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.