Fredholm alternative

In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.

If V is an n-dimensional vector space and ${\displaystyle T:V\to V}$ is a linear transformation, then exactly one of the following holds:

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:

In other words, A x = b has a solution ${\displaystyle (\mathbf {b} \in \operatorname {rng} (A))}$ if and only if for any y s.t. A^Ty = 0, y^Tb = 0 ${\displaystyle (\mathbf {b} \in \ker(A^{T})^{\bot })}$.

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