Linear Algebra: Vector Spaces; Dependent/Independent

[mateomy]

Decide if the indicated set of functions are independent or dependent, and prove your answer.

$$\left\{cos^2(x),sin^2(x),sin(2x)\right\}$$

This linear algebra course is killing me. It's much more abstract than I thought it would be. I realize this problem isn't exactly that, but I am so overwhelmingly frustrated with this class.

Dependence/Indepence is determined if one of the vectors is/is not the zero vector. But for functions how do I interpret this? Do I set it up like an equation and set it to the zero vector? AAAHHHH! So. Completely. Lost. I've read this section twice now and I feel I have nothing to show for it.

 

[deluks917]

You need to determine whether:

asin²(x) + bcos²(x) +csin(2x) = 0

has any solution except a=b=c=0.

Basically can you write any of the three as linear cominqtions of the others. Do you know any realtions between sin², cos² and sin(2x)?

 

[mateomy]

I realize that I can't do that, but how do I prove it? I was thinking about constructing a matrix but that seems ridiculous.

 

[deluks917]

span(u,v) = span(u+v,u-v)

You can replace sin^2 and cos^2 with sin^2 + cos^2 and sin^2 -cos^2. Can you prove the resulting 3 functions are independent.