Proving Linear Independence in a Subset of Trigonometric Functions

[Auron87]

I'm stuck on a question in linear algebra, it reads "Show that the subset S={cos mx, sin nx: m between 0 and infinity, n between 1 and infinity} is linearly independent.

I really just don't know where to start. I've seen a similar question which was just sin (nx) and the lecturer integrated sin(px)sin(qx) between -pi and pi but I just don't see why he did that or anything.

Any starting help would be much appreciated, thanks.

 

[Galileo]

I really just don't know where to start. I've seen a similar question which was just sin (nx) and the lecturer integrated sin(px)sin(qx) between -pi and pi but I just don't see why he did that or anything.

My guess is he showed that sin(px) and sin(qx) are orthogonal if p/=q. If two nonzero vectors are orthogonal then they are linearly independent.

 

[HallsofIvy]

The definition of "linearly independent", applied here would be that
$a_0+ a_1cos(x)+ b_1sin(x)+ a_2cos(2x)+ b_2sin(2x)+ ...= 0$ only when each $a_i$ and $b_i$ is 0. What would you get if you multiply that sum by sin(nx) or cos(nx), for all n, and integrate?