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Thread: Bilinear Forms

  1. #1
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    Bilinear Forms

    Here's the question:


    Part (a)

    I think to show bilinearity, I must prove these two conditions (left linear form & right linear form):

    $\displaystyle \int^{b}_{a} f(\lambda u_1+ \mu u_2,v)g(\lambda u_1+ \mu u_2,v)$

    $\displaystyle \int^{b}_{a} f(u, \lambda v_1 + \mu v_2)g(u, \lambda v_1 + \mu v_2)$

    Can anyone show me how we can prove this, so I can get started? Thanks.

    Edit: $\displaystyle \forall \lambda, \mu \in IR$ and $\displaystyle u_i , v_i \in C[a,b]$
    Last edited by Roam; Oct 4th 2009 at 07:49 PM.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    I don't understand why the functions $\displaystyle f,g$ in your post are functions of two variables. You want to show that the operator $\displaystyle I$ is a bilinear form, not the functions $\displaystyle f,g$.

    You want to show that

    1) $\displaystyle I(f+g,h)=I(f,h)+I(g,h)$

    2) $\displaystyle I(f,h+g)=I(f,h)+I(f,g)$

    3) $\displaystyle I(af,g)=I(f,ag)=aI(f,g)$ for any $\displaystyle a \in \mathbb{R}$.
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  3. #3
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    Quote Originally Posted by Bruno J. View Post
    I don't understand why the functions $\displaystyle f,g$ in your post are functions of two variables. You want to show that the operator $\displaystyle I$ is a bilinear form, not the functions $\displaystyle f,g$.

    You want to show that

    1) $\displaystyle I(f+g,h)=I(f,h)+I(g,h)$

    2) $\displaystyle I(f,h+g)=I(f,h)+I(f,g)$

    3) $\displaystyle I(af,g)=I(f,ag)=aI(f,g)$ for any $\displaystyle a \in \mathbb{R}$.
    Ah! I see what you mean! But where did the "h" come form?

    I think this is what I need to prove:

    $\displaystyle I(\lambda f_1 + \mu f_2, g) = \lambda I (f_1 , g) + \mu I (f_2 , g)$

    $\displaystyle
    I(f, \lambda g_1 + \mu g_2) = \lambda I (f, g_1) + \mu I (f , g_2)
    $

    Could you give me any clues on how to prove these?
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