Parity, Set of functions

Question

Given linearly independent $\{f_n\}_{n=1}^N$ how can we form $n$ linearly independent functions $\{F_n\}_{n=1}^N$ such that each $F_n$ is either an even or odd function? Thanks.

Answer 1

Hint: First define the related functions $g_n(x)=f_n(x) + f_n(-x)$ and $h_n(x) = f_n(x) - f_n(-x)$, so that each $g_n$ is even and each $h_n$ is odd. Then linear combinations of $g_n$ are always even, and linear combinations of $h_n$ are always odd.