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Math Help - Determine a basis

  1. #1
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    Determine a basis

    Let S be the vector space consisting of the set of all linear combinations of the functions
    f1(x)= e^(x)
    f2(x)= e^(-x)
    f3(x)= sinh(x)

    Determine a basis for S, and hence, find dim[S].
    f3(x) = 1/2(f1(x) - f2(x)) so we know that f3(x) is linearly dependent on f1,f2.

    Now, we consider f1(x),f2(x)
    c1*f1(x) + c2*f2(x) = 0

    c1 = c2 = 0. f1 and f2 are Linearly independent and are a basis. Dim[S] = 2.

    Is my answer correct?
    Thanks
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  2. #2
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    Quote Originally Posted by bulldog106 View Post
    f3(x) = 1/2(f1(x) - f2(x)) so we know that f3(x) is linearly dependent on f1,f2.

    Now, we consider f1(x),f2(x)
    c1*f1(x) + c2*f2(x) = 0

    c1 = c2 = 0. f1 and f2 are Linearly independent and are a basis. Dim[S] = 2.

    Is my answer correct?
    Thanks
    yes, it's correct. although you need to prove that c_1=c_2=0.
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  3. #3
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    Quote Originally Posted by bulldog106 View Post
    f3(x) = 1/2(f1(x) - f2(x)) so we know that f3(x) is linearly dependent on f1,f2.

    Now, we consider f1(x),f2(x)
    c1*f1(x) + c2*f2(x) = 0

    c1 = c2 = 0. f1 and f2 are Linearly independent and are a basis. Dim[S] = 2.

    Is my answer correct?
    Thanks
    As NonCommAlg said, you need to prove that c1= c2= 0, not just assert it!

    If c1f1(x)+ c2f(x)= c1e^{x}+ c2e^{-x}= 0 for all x, then it is a constant and so its derivative, c1e^x- c2e^{-x}= 0 for all x also. Take x= 0 in both equations.
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