I came across this theorem while reading basic algebraic properties of Petri Nets and I wanted to know how many of you are acquainted with it. Please post in the comments!
One of the most useful theorems in applied mathematics is the Fredholm Alternative. However, because the theorem has several parts and gets expressed in different ways, many people don’t know why it has “alternative” in the name. For them, the theorem is a means of constructing solvability conditions for linear equations used in perturbation theory.
The Fredholm Alternative Theorem can be easily understood if you consider solutions to the matrix equation $latex A v = b$, for a matrix $latex A$ and vectors $latex v$ and $latex b$. Everything that applies to matrices can then be generalized to infinite dimensional linear operators that occur in differential or integral equations. The theorem is: Exactly one of the two following alternatives hold
- $latex A v = b$ has one and only one solution
- $latex A^* w = 0$ has a nontrivial solution
where $latex A^*$ is the transpose or adjoint of A. …
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