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Math Help - One Linear Algebra Problem (Dual Spaces)

  1. #1
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    One Linear Algebra Problem (Dual Spaces)




    Thank You.
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  2. #2
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    Quote Originally Posted by eyke View Post



    Thank You.

    What's the problem? You need three polynomials p_i(x)\,,\,i=1,2,3 s.t. f_i(p_j(x))=\delta_{ij}.
    For example, you need that \int_0^1p_1(x)\,dx=1\,,\,\,\int^2_0p_1(x)\,dx=0\,,  \,\int^{-1}_0p_1(x)\,dx=0 . Putting p_1(x)=ax^2+bx+c , this means :

    1=\int^1_0(ax^2+bx+c)dx=\frac{a}{3}+\frac{b}{2}+c

    0=\int^2_0(ax^2+bx+c)dx=\frac{8a}{3}+2b+2c

    0=\int^{-1}_0(ax^2+bx+c)dx=-\frac{a}{3}+\frac{b}{2}+c

    Solving the resulting linear system we get p_1(x)=\frac{3}{2}x^2-5x+3 . Do the same with the other two polynomials.

    Tonio
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