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Thread: 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 $\displaystyle p_i(x)\,,\,i=1,2,3$ s.t. $\displaystyle f_i(p_j(x))=\delta_{ij}$.
    For example, you need that $\displaystyle \int_0^1p_1(x)\,dx=1\,,\,\,\int^2_0p_1(x)\,dx=0\,, \,\int^{-1}_0p_1(x)\,dx=0$ . Putting $\displaystyle p_1(x)=ax^2+bx+c$ , this means :

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

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

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

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

    Tonio
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