Determine whether the following vectors are linearly independent.

Question

I have the following question:

"Examine whether the following vectors are linearly dependent or independent. For linear dependency, write a vector as linear combination of the others. For linear independence, check if the vectors form a basis for the respective vector space."

$f_1, f_2, f_3 \in C(\mathbb{R})$ (the vector space of the continuous functions $\mathbb{R} \rightarrow \mathbb{R}$) with $f_1(x) = \sin^2{x}, \ f_2(x) = cos^2(x) \ \& f_3(x) = \cos{2x}.$

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Progress so far:

Check for linear independence: Setting $x = \pi(n - \frac{1}{2}) $ with $n \in \mathbb{z}$ & $a_1 = a_2 = a_3 = 1$ demonstrates that $a_1 f_1(x) + a_2 f_2(x) + a_3 f_3(x) = 0$ is not true only for the case that $a = 0$.

$\therefore$ I need to write $f_1, f_2 \ \& \ f_3$ as linear combinations of each other, at which point i get confused, as the value of a linear combination the functions depends not only on the values $a_1, \ a_2 \ \& \ a_3$, but also on the variable $x$.

Can anyone help me from here?

Thanks in advance.

Answer 1

Hint. Note that by the addition formula, $$\cos(2x)=\cos(x+x)=\cos(x)\cdot \cos(x)-\sin(x)\cdot \sin(x).$$

Answer 2

$cos(2x)=2cos^2(x)-1=2cos^2(x)-(sin^2(x)+cos^2(x))$ you deduce that $f_3=2f_2-f_1-f_2$ so they are not independent.

Answer 3

for $x=0$ we get: $b+c=0$ for $x=\frac{\pi}{2}$ we get: $a-c=0$ and for $x=\pi$: $b+c=0$