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Math Help - determinant of linear maps

  1. #1
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    determinant of linear maps

    Let V be the vector space of functions which are linear combinations of
    1, cos x, sin x, cos 2x, sin 2x, . . . , cos(N x), sin(N x).

    Let φ : V → V be the linear map

    φ(f ) =∫ f (u) du. between x and 0


    Calculate det φ.


    I am not sure what basis to use for V when solving this. Any help would be appreciated
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: determinant of linear maps

    Quote Originally Posted by beano2913 View Post
    Let V be the vector space of functions which are linear combinations of 1, cos x, sin x, cos 2x, sin 2x, . . . , cos(N x), sin(N x). Let φ : V → V be the linear map φ(f ) =∫ f (u) du. between x and 0 Calculate det φ.
    \varphi is not well defined. Consider for example n=1 then, V=\mathcal{L}\{1,\cos x,\sin x\} and \varphi (1)=\int_x^01\;du=-x\not \in V .
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