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Thread: coordinate

  1. #1
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    coordinate

    im very lost with transition matrix..thought i understand it alr but turns out i haven..

    if i define T to be the transition matrix from S to S'
    then let (v)_s be the coordinate vector wrt to S...

    then its right of me to say that the coord vector wrt to S' is T(v)_s right?


    let S= {v_1,...,v_n}, S' = {v'_1,...,v'_n} be the basis for the vector space V
    then in my notes, it states that (v_1,...,v_n) = (v'_1,...,v'_n)T. but i thought it should be
    T(v_1,...,v_n) = (v'_1,...,v'_n) since T is from S to S'?

    i dont get what it means when my prof puts the T behind the vectors.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Perhaps this example for a vector space $\displaystyle E$ with $\displaystyle \dim E=2$ may help you:

    Let $\displaystyle B=\left\{{u_1,u_2}\right\},\;B'=\left\{{u'_1,u'_2} \right\}$ be two basis of $\displaystyle E$.

    Suppose:

    $\displaystyle \left\{ \begin{array}{c} u'_1=3u_1-2u_2 \\ u'_2=4u_1+5u_2 \end{array}\right$

    Denote:

    $\displaystyle {}^t(x_1,x_2)$ : coordinates of $\displaystyle x\in E $ in respect to $\displaystyle B $

    $\displaystyle {}^t(x'_1,x'_2)$ coordinates of $\displaystyle x\in E $ in respect to $\displaystyle B' $

    Then:

    $\displaystyle x=x'_1u'_1+x'_2u'_2=x_1u_1+x_2u_2$

    Identifying coordinates on $\displaystyle B$, we easily obtain:

    $\displaystyle \left({\begin{array}{ccc}{x_1}\\{x_2}\end{array}\r ight)=\begin{pmatrix}{3}&{4}\\{-2}&{5}\end{pmatrix}\left({\begin{array}{ccc}{x'_1} \\{x'_2}\end{array}\right)$

    Or equivalently:

    $\displaystyle \left({x_1},\;{x_2}\right)=\left({x'_1},\;{x'_2}\r ight)\begin{pmatrix}{3}&{-2}\\{4}&{5}\end{pmatrix}=(x_1',x_2')T$

    It's easy to generalize for $\displaystyle \dim E=n$ .

    Regards.
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