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Eigenfunctions of a linear operator

Suppose that O1 and O2 are eigenfunctions of a linear operator A with eigenvalues a1 and a2, respectively, and that we construct a linear combination Y=C1O1 + C2O2. Under what conditions does Y became an eigenfunction of A? Show reasoning.

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In any relationship of the type AF = aF, where F is a function, A is an operator and a is a number, F is said to be an eigenfunction of the operator A with eigenvalue equal to a. ...