Are the vectors
$\displaystyle
6 , 3\sin^2 x , 2\cos^2 x
$ in F(-infinity,infinity) the space of all functions from R to R linearly dependent or linearly independent?
Been a while since I've done this, but if I recall (and it makes sense, given the terminology), if you can write one of the vectors as a linear combination of the other vectors, then they're not all linearly independent. If this assumption is incorrect, well, throw this post away.
Note that we can rewrite $\displaystyle 3sin^{2}x$ as $\displaystyle 3(1-cos^{2}x) = 3 - 3cos^2{x}$.
That's certainly a combination of the others:
3 is simply half the "6" vector, and the other part is 1.5 times the cosine-squared vector.
So I vote that they are not linearly independent.
I have answered this question in http://www.mathhelpforum.com/math-he...ly+independent. You could have asked me if you didnt understand, rather than editing the post