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Math Help - Invertibility of a Function

  1. #1
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    Invertibility of a Function

    This question Im really lost on, a bit of guidance would be great!

    Let J_n be the R-vector space with basis B = \{1,cos(x),sin(x),...,cos(nx),sin(nx)\}. For a fixed positive real number a, define D \in Hom(J_n,J_n) by D(f) = f'' + a^2f. For which a is D invertible?
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  2. #2
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    Quote Originally Posted by joe909 View Post
    This question Im really lost on, a bit of guidance would be great!

    Let J_n be the R-vector space with basis B = \{1,cos(x),sin(x),...,cos(nx),sin(nx)\}. For a fixed positive real number a, define D \in Hom(J_n,J_n) by D(f) = f'' + a^2f. For which a is D invertible?

    Check the efect of the transformation on the given basis' elements:

    D(1)=a^2\Longrightarrow so far it must be a\neq 0

    D(\cos x)=-\cos x +a^2\cos x=\cos x(a^2-1)\Longrightarrow it also has to be a\neq \pm 1

    D(\cos 2x)=-4\cos x+a^2\cos x =\cos 2x(a^2-4)\Longrightarrow it also has to be a\neq \pm 2 ....

    Tonio
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  3. #3
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    Thanks, I get it, if  D(x) = 0 for x \neq 0 then it is no longer one to one. I think I just got caught up in wording of the question.
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