Question

LetVbethesetofallfunctionsthataretwicedifferentiableinRand S={cosx,sinx,xcosx,xsinx}. a)CheckthatSisalinearlyindependentsetoverR.(Hint:Considertheequation a0cosx+a1sinx+a2xcosx+a3xsinx. Putx=0,π,π 2 ,π 4 ,etc.andsolveforai.)

Answer 1

Answer on Question#54663, Math / Linear Algebra

For set of functions $\text{sinx}, \text{cosx}, \text{xcosx}, \text{xsinx}$ to be linearly independent, $a_0 \cos x + a_1 \sin x + a_2 \cos x + a_3 \sin x = 0$ only when $a_0 = 0, a_1 = 0, a_2 = 0, a_3 = 0$ .

Let us evaluate $a_0 \cos x + a_1 \sin x + a_2 \cos x + a_3 \sin x = 0$ for $x = 0, x = \pi, x = \frac{\pi}{2}, x = \frac{\pi}{4}$ . Obtain:

$$x = 0: a _ {0} = 0$$ $$x = \pi : - \pi a _ {2} = 0$$ $$x = \frac {\pi}{2}: a _ {1} + \frac {\pi}{2} a _ {3} = 0$$ $$x = \frac {\pi}{4}: \frac {a _ {1}}{\sqrt {2}} + \frac {\pi}{4 \sqrt {2}} a _ {3} = 0$$

From first two equations, $a_0 = 0, a_2 = 0$ , and substituting third equation into fourth, obtain $a_1 = 0$ and $a_3 = 0$ .

Therefore, the set of functions $\text{sinx}, \text{cosx}, \text{xcosx}, \text{xsinx}$ are linearly independent.

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