Linear dependent functions

Question

V is a linear space of all functions $\ \mathbb R \to \mathbb R $ and $\ f,g,h \in V $

$\ f(x) = x\cos x $ , $\ g(x) = \cos x $, $\ h(x) = \sin x $

Need to see if those functions are dependent.

$\ 0 = \alpha_1 x\cos x + \alpha_2\cos x + \alpha_3 \sin x $

lets say $\ t= \cos x , s = \sin x $

$\ \alpha_1 x t + \alpha_2 t + \alpha_3 s $ = 0

$\ t(\alpha_1x + \alpha_2) + s \alpha_3 = 0 $

Not sure how do i go on from here? and I haven't been taught the Wronskian.

Answer 1

Just working out Alan's hint: $$\ 0 \equiv \alpha_1 x\cos x + \alpha_2\cos x + \alpha_3 \sin x$$ Set

• $x= 0 \Rightarrow \alpha_2 = 0$
• $x= \pi \stackrel{\alpha_2 = 0}{\Rightarrow} -\alpha_1 \pi = 0 \Rightarrow\alpha_1 = 0$
• $x=\frac{\pi}{2}\stackrel{\alpha_1=0,\alpha_2 = 0}{\Rightarrow}\alpha_3 = 0$

It follows that the three functions are linearly independent, as the coefficients $\alpha_1, \alpha_2, \alpha_3$ have to be zero.

Answer 2

Hint. If your first equation is true, it is true for all values of x. Try specific values of x, that will give you multiple equations that are linear in your coefficients. Show they are inconsistent.