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Math Help - matrix of inner product on span(s) with respect to a ordered basis

  1. #1
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    matrix of inner product on span(s) with respect to a ordered basis

    so i have no idea how to approach this second part of the problem. for the first part, i've figured out that the vectors in the set s are indeed linearly independent. but as far as the second part is concerned, i have no idea what the question even is.
    any help is appreciated.
    thanks !!

    Let V be the vector space of infinitely differentiable functions on R.
    1. prove that the set S={e^x, x, x^2} is linearly independent
    2. Find the matrix of the inner product <f,g> = integrate from -1 to 1 of f(x)g(x)dx on Span(S) with respect to the ordered basis {e^x, x, x^2}.

    heres an image
    matrix of inner product on span(s) with respect to a ordered basis-untitled.jpg
    Last edited by bombardior; May 6th 2010 at 10:03 PM.
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  2. #2
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    Quote Originally Posted by bombardior View Post
    so i have no idea how to approach this second part of the problem. for the first part, i've figured out that the vectors in the set s are indeed linearly independent. but as far as the second part is concerned, i have no idea what the question even is.
    any help is appreciated.
    thanks !!

    Let V be the vector space of infinitely differentiable functions on R.
    1. prove that the set S={e^x, x, x^2} is linearly independent
    2. Find the matrix of the inner product <f,g> = integrate from -1 to 1 of f(x)g(x)dx on Span(S) with respect to the ordered basis {e^x, x, x^2}.

    heres an image
    Click image for larger version. 

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    "Linearly independent" means that if a e^x+ bx+ cx^2= 0 for all x, then a= b= c= 0. That should be easy to prove.

    An inner product, <u, v>, can be written as a matrix by representing u and v as "row" and "column" matrices in <u, v>= uAv. Here, the space is 3 dimensional so A must be a 3 by 3 matrix.

    For example, the first basis vector is e^x and the inner product of that with itself would be represented as \begin{bmatrix}1 & 0 & 0\end{bmatrix}\begin{bmatrix}a_{11} & a_{12} & a_{31} \\ a_21 & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}= a_{11} and that must be equal <e^x, e^x>= \int_{-1}^1 (e^x)^2 dx = \int_{-1}^1 e^{2x} dx.

    In general, [tex]a_{ij}[tex] is the inner product of the "ith" basis vector with the "jth" basis vector.
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  3. #3
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    so if i understand correctly, does this just mean i've trying to use a matrix to represent the integral instead of using the inner product definition?
    how does the ordered basis come into play?
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