The Fredholm Alternative is a principle in functional analysis that deals with the solutions of certain linear equations. It states that for a linear operator, if the homogeneous equation has a non-trivial solution, then the inhomogeneous equation has a solution if and only if the right-hand side is orthogonal to the kernel of the operator. This concept is crucial in understanding the behavior of solutions to differential equations and integral equations. This principle is particularly important in the study of Fredholm operators, which are bounded linear operators with a finite-dimensional kernel and cokernel. The Fredholm Alternative helps determine the existence and uniqueness of solutions, providing a framework for analyzing various mathematical and physical problems, including those in quantum mechanics and engineering.