Chapter 4.2, Problem 45E

Mass-Spring System The mass in a mass-spring system (see figure) is pulled downward and then released, causing the system to oscillate according to $x\left(t\right)=a_1\sin \omega t+a_2\cos \omega t$ where $x$ is the displacement at time $t,a_1$ and $a_2$ are arbitrary constant, and $\omega$ is a fixed constant. Show that the set of all functions $x\left(t\right)$ is a vector space.