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Math Help - Orthogonal Basis for F

  1. #1
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    Orthogonal Basis for F

    Let F be the set of functions  f: \mathbb{R}^{+} \rightarrow \mathbb{R} of the form  f(x) = ax + log(x^b) . Define an inner product on F by  <f,g> = \int^e_1 f(t) g(t)\, dt

    Find an othogonal basis for F
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  2. #2
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    I think you have to find 2 functions, say h(t) and j(t) such that \int^e_1 h(t) j(t)\, dt=0 and of course that h(x) = cx + log(x^d) and j(x) = mx + log(x^k).
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  3. #3
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    Quote Originally Posted by Jimmy_W View Post
    Let F be the set of functions  f: \mathbb{R}^{+} \rightarrow \mathbb{R} of the form  f(x) = ax + log(x^b) . Define an inner product on F by  <f,g> = \int^e_1 f(t) g(t)\, dt

    Find an othogonal basis for F
    since F=\{ax+b \log x: \ \ a,b \in \mathbb{R} \}, we have F=\text{span} \{x, \log x \}. to find an orthogonal basis, it's easier to use the Gram-Schmidt process:

    put u=x and v= \log x - \frac{<x, \log x>}{<x.x>} \ x. find v and then \{u,v \} will be an orthogonal basis for F.
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  4. #4
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    I have become lost.....but in trying to continue with what you provided me with I have come up with the following:

     v= \log x - \frac{<x, \log x>}{<x,x>} \ x.

     = \log x - \frac{\int_0^1 t \log t \ dt}{\int_0^1 t^2 \ dt} (\log x)

     = \log x - \frac{\int_0^1 t \log t \ dt}{1/3} (\log x)

    Am I barking up the wrong tree here? If so, what do I need to do because Ive been trying to figure this out for a while. If not, where do I go from here?
    Thanks
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  5. #5
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    Quote Originally Posted by Jimmy_W View Post
    I have become lost.....but in trying to continue with what you provided me with I have come up with the following:

     v= \log x - \frac{<x, \log x>}{<x,x>} \ x.

     = \log x - \frac{\int_0^1 t \log t \ dt}{\int_0^1 t^2 \ dt} \color{red} (\log x)

     = \log x - \frac{\int_0^1 t \log t \ dt}{1/3} \color{red} (\log x)

    Am I barking up the wrong tree here? If so, what do I need to do because Ive been trying to figure this out for a while. If not, where do I go from here?
    Thanks
    where did those \color{red} \log x come from? you need to change them to x. another thing is that the limits of your integrals are 1 and e not 0 and 1. (see your question again!)

    finally to find \int_1^e t \log t \ dt just use integration by parts: \log t = u, \ tdt=dv.
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  6. #6
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    f(t)=span(\ln t, t)=span(\vec{v}_1, \vec{v}_2),then\ \vec{u}_1=\frac{\vec{v}_1}{\|\vec{v}_1\|}
    \|\vec{v}_1\|^2=<\vec{v}_1,\vec{v}_1>=<\ln t,\ln t>=\int^e_1 \ln ^2(t) dt=e-2,\vec{u}_1=\frac{\ln t}{\sqrt{e-2}}

    \vec{v}_{2}^{\perp}=\vec{v}_{2}-\vec{v}_{2}^{\parallel}=\vec{v}_{2}-proj_{\vec{u}_1}\vec{v}_{2}=\vec{v}_{2}-(\vec{u}_1\cdot \vec{v}_2)\vec{u}_1=\vec{v}_{2}-<\vec{u}_1,\vec{v}_2>\vec{u}_1=t-\frac{(e^2+1)\ln t}{4(e-2)}

    We solve orthogonal basis \mathbb{B}=(\ln t,t-\frac{(e^2+1)\ln t}{4(e-2)})\ or\ (\frac{\ln t}{\sqrt{e-2}},t-\frac{(e^2+1)\ln t}{4(e-2)})

    Orthnormal basis is (\vec{u}_1,\vec{u}_2)=(\vec{u}_1, \frac{\vec{v}_{2}^{\perp}}{\|\vec{v}_{2}^{\perp}\|  })

    You could choice \vec{v}_1=t, there are no unified solutions.
    Last edited by math2009; June 1st 2009 at 09:45 PM.
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