Answer to Question #326597 in Differential Equations for Allen

Question

 Show that the functions 1, cos2 x, sin2 x are linearly dependent .

Answer 1

Assume that there are complex numbers $a_1,a_2,a_3$ not all equal to zero satisfying: $a_1+a_2cos\,2x+a_3sin\,2x=0$. Choose different values of $x:$ $x=0,x=\frac{\pi}{4},x=\frac{\pi}{3}$. We get the following equations: $a_1+a_2=0,\,\, a_1+a_3=0=0\,\,a_1-\frac12a_2+a_3\frac{\sqrt{3}}{2}=0$. From first and second equations we get: $a_2=a_3=-a_1$ . Third equation implies: $a_3=0$. Thus, all numbers are zero. We came to contradiction and received that all functions are linearly independent.