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 as .
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.