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Math Help - Linear Map Problem

  1. #1
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    Linear Map Problem

    I Just started the course. I need more example problem.

    I know how to find a formula for f, but the question 1 blew, given what is satisfied, how do i compute the f ? What is the meaning of that?

    what is a linea transformation? my instructor just threw problems, hardly explains..






    http://www.Photo-Host.org/img/589464math207.png
    Last edited by CaptainBlack; September 16th 2008 at 11:30 PM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by yzc717 View Post
    I Just started the course. I need more example problem.

    I know how to find a formula for f, but the question 1 blew, given what is satisfied, how do i compute the f ? What is the meaning of that?

    what is a linea transformation? my instructor just threw problems, hardly explains..
    I think you need to provide more information (I can't see your image file as photo hosting sites are blocked on this computer), but:

    A linear transformation f on a vector space S over a field F (usualy \mathbb{R} or \mathbb{C}) to another vector space over the same field, is a transformation such that:

     <br />
\forall u,v \in S, \alpha, \beta \in F;\ \ f(\alpha u + \beta v) =\alpha f(u)+\beta f(v)<br />


    RonL
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  3. #3
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    can u see the photo now?


    how do i prove the problem 4?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by yzc717 View Post
    can u see the photo now?


    how do i prove the problem 4?
    Yes, I can now see the picture.

    The i -th diagonal element of AB is:

    d_i=\sum_j A_{i,j}B_{j,i}

    Hence the trace of AB is:

    tr(AB)=\sum_i d_i=\sum_i \sum_j A_{i,j}B_{j,i}

    Now these are finite sums so the order of summation can be reversed, so:

    tr(AB)=\sum_j \sum_i A_{i,j}B_{j,i}=\sum_j \sum_i B_{j,i}A_{i,j}=tr(BA)

    RonL
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