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Math Help - How to prove 2 functions are linearly independent

  1. #1
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    How to prove 2 functions are linearly independent

    Hi.

    i need to prove that \sin x and \cos x are linearly independent.
    what steps should i start with?

    and how do i prove functions' linearly independent in general?

    thanks in advanced!
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  2. #2
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    Re: How to prove 2 functions are linearly independent

    Suppose a\sin x+b\cos x=0 for all x. Substitute x=0 and x=\pi/2.
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    Re: How to prove 2 functions are linearly independent

    if x=0 then b=0, and if x=\frac{\pi}{2} then a=0, but what about every other x?
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    Re: How to prove 2 functions are linearly independent

    Can you formulate what it means for sin(x) and cos(x) to be linearly independent?
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    Re: How to prove 2 functions are linearly independent

    that's the thing, i'm not sure.
    i only know how to check for liniar dependence when it is vectors from  \mathbb{R}^n
    and then if there exist scalars a_0, a_1, .... a_n \in \mathbb{R} not all 0, so that a_0\cdot v_0+a_1\cdot v_1 .... a_n\cdot v_n=0 then the vectors linear dependent.
    but here with this x - i can't get the hang of it.
    i get that i need to show that there aren't A and B not all 0 so that A\cdot \sin x+B\cdot \cos x=0, but which x are we talking about here?
    can you give me an example of two linear dependent functions? mayby that will help.
    Last edited by Stormey; December 26th 2012 at 12:51 PM.
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  6. #6
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    Re: How to prove 2 functions are linearly independent

    The definition of linearly independent functions is the same as for vectors, but the issue is in the definition of +, *, = and 0. Here + denotes pointwise addition of functions, multiplication is pointwise multiplication of a function by a constant, and equality is equality of functions, not numbers. When we say that f = g, it means that f(x) = g(x) for all x (this definition of function equality is sometimes called extensional because it depends only on the functions' output and not their algorithms). Finally, 0 is the function that returns the number 0 for all arguments. So, in proving that sin(x) and cos(x) and linearly independent, we assume that there exist a and b such that a * sin(x) + b * cos(x) = 0 for all x. Then we need to show that a = b = 0, which you have done.

    An example of two linearly dependent functions is \sin(x+\pi/3) and \sin x+\sqrt{3}\cos x. As with vectors, if f and g are linearly dependent, then f = a * g for some constant a.
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    Re: How to prove 2 functions are linearly independent

    i get that i need to show that there aren't A and B not all 0 so that , but which x are we talking about here?
    emakarov answered this in his very first response: when we talk about two functions, f and g, being linearly independent we mean that the only a and b such that af(x)+ bg(x)= 0 for all x are a= b= 0. In particular, if a sin(x)+ bcos(x)= 0 for all x, then, for x= [itex]\pi/2[/itex], we have a(1)+ b(0)= a= 0. Now we know that bcos(x)= 0 for all x. Either b= 0 or cos(x)= 0 for all x.
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  8. #8
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    Re: How to prove 2 functions are linearly independent

    OK thanks for clearing that out for me.
    that was very helpful!
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