How do you prove whether the set of functions $\{1,\sin 2x, \sin^2 x\}$ is linearly independent (in the vector space of all continuous functions)? From the problem set of an ODE course:
How do you prove whether the set of functions $\{1,\sin 2x, \sin^2 x\}$ is linearly independent (in the vector space of all continuous functions)?
I understand that by definition, the functions are linearly dependent iff the only constants $a$, $b$, and $c$ that satisfy $a+b\sin 2x + c\sin^2 x = $ are $a=b=c=0$, but how do you (concisely) show whether this is the case?
 A: Suppose $f(x)=\alpha_1 + \alpha_2 \sin (2x) + \alpha_3 \sin^2 x = 0$ for all $x$.
Since $f(0) = \alpha_1$, we see that $\alpha_1 = 0$.
Since $f'(0) = 2 \alpha_2$, we see that $\alpha_2 = 0$.
Since $f({\pi \over 2}) = \alpha_3$, we see that $\alpha_3 = 0$.
Alternatively, the first three terms of the Taylor series are
$f(x) = \alpha_1 + 2 \alpha_2 x + \alpha_3 x^2 + \cdots$,
from which we can read off  $\alpha_1 = \alpha_2 = \alpha_3 = 0$.
A: EDIT: Now that the OP has changed his question, my answer looks silly. He didn't previously have any info about scalars all equal to zero. Here is my original response:
A set of vectors $\{v_1,v_2,v_3\}$ is linearly independent if and only if the only scalars $a_i$ such that $a_1v_1+a_2v_2+a_3v_3=0$ are $a_1=a_2=a_3=0$.
For your space, we have $a_1+a_2\sin 2x+a_3\sin^2 x=0$. Are there scalars (coefficients) $a_1, a_2, a_3$ other than $a_1=a_2=a_3=0$ such that this is always true? I'll leave it to you to show that there are not. Hence, your set is linearly independent.
