# How to prove Linear Independence in the below 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.

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 $$a = 0$$.
• you could set $x =\pi/2$ to show $a = 0$, since $a \sin(\pi/2) + b \cos(\pi/2) =0 \Rightarrow a = 0$ – Physor Sep 23 at 21:23