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 $a = 0$.

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  • $\begingroup$ you could set $x =\pi/2$ to show $a = 0$, since $a \sin(\pi/2) + b \cos(\pi/2) =0 \Rightarrow a = 0$ $\endgroup$ – Physor Sep 23 at 21:23
  • $\begingroup$ I was hoping somebody would explain using Wronksian and then i would ask about it. $\endgroup$ – Harsh Vardhan Singh Sep 23 at 21:32

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