Fredholm Alternative - Linear Algebra

Linear Algebra

If V is an n-dimensional vector space and is a linear transformation, then exactly one of the following holds:

1. For each vector v in V there is a vector u in V so that . In other words: T is surjective (and so also bijective, since V is finite-dimensional).
2. .

A more elementary formulation, in terms of matrices, is as follows. Given an m×n matrix A and a m×1 column vector b, exactly one of the following must hold:

1. Either: A x = b has a solution x
2. Or: AT y = 0 has a solution y with yTb ≠ 0.

In other words, A x = b has a solution if and only if for any y s.t. AT y = 0, yTb = 0 .