Is the vector space {sin(x), sin(2x), sin(3x)} linearly independent or linearly dependent?

Question

My guess is linearly independent, but i keep second guessing myself. How do I prove the answer?

Answer 1

set a linear combination of your 3 vectors =0 and prove that the numbers have to be =0:
A sin(x) + B sin(2x) + C sin(3x) = 0
i.e. A,B,C are numbers such that the linear combination is the constant zero.
now take x=pi/2,
then
0= A sin(pi/2) + B sin(2pi/2) + C sin(3pi/2)
= A - C,
so A=C.
Now x=pi/4
0= A sin(pi/4) + B sin(2pi/4) + A sin(3pi/4)
= A/sqrt(2) + B + A/sqrt(2)
= 2 A/sqrt(2) +B
= sqrt(2) A + B, so B= -sqrt(2) A
Now, x = 3pi/4
0= A sin(3pi/4) + B sin(2(3)pi/4) + A sin(9pi/4)
= A/sqrt(2) - B + A/sqrt(2)
=sqrt(2) A - B, but then B = sqrt(2) A
which means that -sqrt(2) A = sqrt(2) A
so A =0, therefore B=0 and C=0
'

Answer 2

You want to be careful of your terminology. To be a vector space, you need to be able to add and multiply by scalars and stay in the vector space. Also, vector spaces are neither dependent nor independent. Certain subsets of vector spaces can be, however.

In your case, the vector space is probably the collection of all continuous functions or some such thing. The *set* {sin(x), sin(2x), sin(3x)} is either dependent or independent. So see if some linear combination can be 0:

Asin(x)+Bsin(2x)+Csin(3x)=0.

This should happen for every value of x. By choosing appropriate values of x, you can get conditions on A,B, and C for this equality to happen. If you can conclude that A=B=C=0, then you know your set is independent. This turns out to be the case.