A linear combination in the context of Fourier series is a sum of whatever basis element and an arbitrary constant
\sum_{i}\alpha_{i}\hat{e}_{i} = a\hat{e}_{1} + b\hat{e}_{2} + c \hat{e}_{3} $$To represent an arbitrary function as a series of constituent basis components or eigenfunctions? you can represent the function as a linear combination of the basis vectors. This is the basis for [[Fourier Series]]'