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Use coordinate vectors to determine whether or not the given set is linearly independent. If it is linearly dependent, express one of the vectors as a linear combination of the others.

The set S, is [tex]\left\{ {2 + x - 3\sin x + \cos x,x + \sin x - 3\cos x,1 - 2x + 3\sin x + \cos x,2 - x - \sin x - \cos x,2 + \sin x - 3\cos x} \right\}[/tex].

So I assume that I start off by letting c_i (i = 1,2,3,4,5) be scalars multiply each of the c_i by each of the elements of S and get an equation which looks something like:

(something) + (something else)x + (another thing)sinx + (something different)cosx = 0.

I'm not really sure how to proceed at this point. One of the examples in my book, with a different set S(with 3 elements), substitutes 3 specific values of x into the equation and gets c_1 = c_2 = c_3 = 0 so that the set is linearly independent.

However, I'm not sure if that is the right method because if I have S = {1, sin^2(x), cos^2(x)} then S is linearly dependent since 1 = (1)cos^2(x) + (1)sin^2(x). But if I substitute x = 0, x = pi/2, x = pi into the equation 1 + sin^2(x) + cos^2(x) = 0 then I get a homegeneous system which only has the trivial solution and my books to suggest that it is enough to conclude from that, the set S is linearly independent(when it is clearly isn't as I just demonstrated before).

More specifically, my book says that the equation(say c_1(1) + c_2(x) + c_3(sinx) = 0 must hold for all values of x so it holds for specific values of x. I just don't know if that's a valid 'method' to use. If it is then I could simply substitute 'convenient' values of x for the question that I included at the beginning of this message to get a simple system of equations.

In short, I'm not sure how to proceed with the question I included at the start of this message. Can someone please help me out?