Ways to build a set of linear independent vectors from an orthonormal basis.

Question

Suppose I have an orthonormal basis $\{{\psi_n}\}_{n \in \mathbb N}$ for a Hilbert space $\cal H$.

Consider the elements $\phi_n = \psi_{n+1}+ a \psi_n, \quad \forall n \in \mathbb N, a \in \mathbb C$

Are $\{\phi_n\}_{n \in \mathbb N}$ linearly independent?

Answer 1

Write out an arbitrary linear combination of $\{ \phi _n \} _n $ and use the fact that $ \{ \psi _n \} _n $ are linearly independent.