Determining Linear Independence: Use Coordinate Vectors

[Benny]

Can someone help me out with the following question?

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 $$\left\{ {2 + x - 3\sin x + \cos x,x + \sin x - 3\cos x,1 - 2x + 3\sin x + \cos x,2 - x - \sin x - \cos x,2 + \sin x - 3\cos x} \right\}$$.

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?

 

[AKG]

You should be able to see immediately that it's linearly dependent. The basis vectors here are 1, x, sin(x), cos(x)... there are only 4 of them while there are 5 elements of your set. However, the questions asks you to use co-ordinates to show this, and then express one in terms of the others. So let your basis be {1, x, sin(x), cos(x)}. You can express the first element of your set with respect to this basis with co-ordinates (2,1,-3,1). Do something similar with the other vectors, and then show linear dependence as you would if you were just dealing with vectors in R⁴.

 

[Muzza]

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).

Are you sure the homogenous system you got had only the trivial solution? Suppose a + b * sin^2(x) + c * cos^2(x) = 0 for all values of x, substituting x = 0, x = pi/2 and x = pi (as per your suggestion) I get

{ a + c = 0
{ a + b = 0
{ a + c = 0

(And yes, a + c = 0 is supposed to be repeated). That system clearly has many solutions, for example (a, b, c) = (1, -1, -1).

 

[Benny]

Muzza - Yeah, you're right. I forgot to square the negative one with the cos^2(x) term.

AKG - That seems right. After reading your response I see that what I wasn't doing in my previous attempts was to choose a basis as a starting point.

Thanks for the help.