How to prove Linear Independence in the below question?

Question

I am solving Linear Algebra and stuck in middle of the problem.

''Let V be the set of all real valued functions y = f(x) satisfying y''+4y=0. Prove that V is a 2-dimensional real vector space.''

I know Basis of the above set is {cos 2x, sin 2x}.How to prove Linear Independence in this set ?

Any help is appreciated.

Answer 1

Suppose $\sin$ and $\cos$ are linearly dependent, that is, there exist constants $a, b$ such that $a \sin(x) + b \cos(x) = 0$ for all $x$. Setting $x = 0$, we get $0 = a \sin(0) + b \cos(0) = a \cdot 0 + b \cdot 1 = b$, hence $b = 0$. But then $a \sin (x) = 0$ for all $x$, which is only true if $a = 0$.