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Math Help - bilinear form

  1. #1
    Junior Member
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    bilinear form

    Hi,

    Can anyone help with the following question:

    Let Vn be the vector space of of polynomials,
    h element of R[x],
    deg(f)<= n.

    For h,k element of Vn - define:

    f(h,k) = integral between 0 and infinity of [ h(x)k(x)(e^-x)] dx

    a) Show that f is symmetric bilinear form

    b) Let B be the basis {1,X, . . . X^n} of Vn. Find [f] w.r.t B.
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  2. #2
    Super Member Rebesques's Avatar
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    You can easily verify that for f:

    (1) f(\cdot,k) is linear.
    (2) f(h,\cdot) is linear.
    (3) f(h,k)=f(k,h) and
    (4) f(th,tk)=t^2f(h,k), \ t\in \mathbb{R}.

    So f is a symmetric bilinear form.

    Its matrix with respect to the basis B will have entries [f]_{ij}=[f(x^i,x^j)]=\left[\int_0^{\infty}x^{i+j}{\rm e}^{-x}dx\right], which you can find by repeated use of the integration by parts formula.
    Last edited by Rebesques; August 26th 2007 at 08:43 AM. Reason: myopia
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