Question

Let V be the vector space of functions that describes the vibration of mas-spring system (Refer \(\left\{ {\sin \omega t,\,\,\cos \omega t} \right\}\)to Exercise 19 in section 4.1.). Find a basis for V.

Short answer

\(\left\{ {\sin \omega t,\,\,\cos \omega t} \right\}\)

Step-by-step solution

Find the set of vectors for V

The equation representing the spring-mass system is \(y\left( t \right) = {c_1}\cos \omega t + {c_2}\sin \omega t\).

So, \(y\left( t \right)\) in V can be expressed as a linear combination of the vectors \(\cos \omega t\) and \(\sin \omega t\).

Check for sine and cosine functions

The sine and cosine functions are not scalar multiplesof each other. So, the sets are linearly independent.

So, the basis of the set of functions is \(\left\{ {\sin \omega t,\,\,\cos \omega t} \right\}\).