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Linear Dependence and Trig

  1. Mar 18, 2009 #1
    1. The problem statement, all variables and given/known data
    show {cos x ,sin x , cos 2x , sin 2x , (cos x − sin x)^2 − 2*sin^2( x)} is not a linearly independent set of real valued functions on the real line R.

    3. The attempt at a solution
    Not linearly independent = linearly dependent?
    So if
    f(x) = cos (x)
    g(x) = sin (x)
    m(x) = cos (2*x) = 1 - 2sin^2(x)
    k(x) = sin (2*x) = 2sin(x)cos(x)
    h(x) = (cos (x) − sin (x))^2 − 2*sin^2(x)

    z(x) = (a*f(x)) + (b*g(x)) + (c*m(x)) + (d*k(x)) + (e*h(x))

    To prove it is linearly dependence we need scalars a,b,c,d,e that work with ANY x that make the equation z(x) equal to zero? Some of the scalars can be zero correct? Just not all of them then it becomes an non trivial answer.

    For instance:

    0=1 * 2sin(x)cos(x) + -1 * (1 - 2sin^2(x) ) + 1 *( cos^2(x) + sin^2(x) - 2sin(x)cos(x) - 2sin^2(x) ) + 0 * cos(x) + 0 * sin(x)
     
  2. jcsd
  3. Mar 18, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, that is correct.

    ?? m, k, and h are not among the functions you give initially. Is this a different example?

    Yes.

     
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