Chapter 8.2, Problem 47P

To determine

To prove:

The set $\left\{e^{ax}\cos bx,x{e}^{ax}\cos bx,x^2{e}^{ax}\cos bx,\mathrm{...},x^m{e}^{ax}\cos bx,e^{ax}\sin bx,x{e}^{ax}\sin bx,x^2{e}^{ax}\sin bx,\mathrm{...},x^m{e}^{ax}\sin bx\right\}$ is linearly independent for all constants $a$ and $b$ on $\left(-\infty ,\infty \right)$.