Linear independency

Question

Why the set S is linearly independent ? Don't we have theorem which indicate that when we have signal set it is linearly independent. For example: (2,5,9,8,6)

The set S={2sin , 2cos , 3} 2 2 x x is linearly dependent (by inspection) : True

Answer 1

We want to know whether or not the set

Is linearly independent. In order to do this, we need to check if any of the 'vectors' in our set can be written as a sum of any of the others. From left to right, we recognize that sine cannot be rewritten with a combination of a scalar (3) and cosine. We also recognize that because of that, cosine cannot be written as a combination of sine and a scalar (3). Finally, there's no way without squaring to rewrite the scalar "3" as a combination of sines and cosines.

I see your confusion though! Keep in mind that sine, and cosine are functions, and the 'theorem' you're quoting refers to a list of just a set of numbers, but do not say anything about linear combinations of functions.