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Math Help - linear algebra change of basis transformation

  1. #1
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    linear algebra change of basis transformation

    Can anyone please help get started with this problem, thanks
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  2. #2
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    Re: linear algebra change of basis transformation

    The transition matrix from $\alpha$ to $\alpha'$ is composed of the coordinates in $\alpha$ of vectors from $\alpha'$ written as columns. I.e., if $\alpha'=(v_1',v_2',v_3')$, we write the coordinates of $v_1'$ in $\alpha$ as the first column, the coordinates of $v_2'$ in $\alpha$ as the second column and so on. Note: some people call this the transition matrix from $\alpha'$ to $\alpha$. Don't have anything to do with them. Kidding.

    Let $C$ be the transition matrix from $\alpha$ to $\alpha'$. If $v=(x,y,z)$ in $\alpha$ and $v=(x',y',z')$ in $\alpha'$, then
    \[
    \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}= C\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}
    \]
    ("old" coordinates are expressed through "new" ones). This makes sense: $v$ is expressed through $v_1',v_2',v_3'$ using coefficients $x_1',x_2',x_3'$, and each of $v_i'$'s is expressed through $v_j$'s using the column vectors of $C$. Ultimately, $v$ is expressed though $v_j$'s. Note that since $C$ converts "primed" coordinates $x_1',x_2',x_3'$ into "non-primed" coordinates $x_1,x_2,x_3$, some people call $C$ the transition matrix from $\alpha'$ to $\alpha$.

    If $C$ is the transition matrix from $\alpha$ to $\alpha'$ in my sense and $A_T$ is the matrix of $T$ in $\alpha$, then the matrix of $T$ in $\alpha'$ is $C^{-1}A_TC$. Indeed, the matrix of an operator converts coordinates of a vector in some basis into the coordinates of the image of that vector in the same basis ("new" coordinates are expressed through "old" ones; compare this with the transition matrix). So if $v=(x_1,x_2,x_3)$ in $\alpha$, then $Tv$ has coordinates $A_T\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}$ in $\alpha$. Meanwhile, let $v=(x_1',x_2',x_3')$ in $\alpha'$. Then $v$ has coordinates $C\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}$ in $\alpha$, $Tv$ has coordinates $A_TC\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}$ in $\alpha$, and finally $Tv$ has coordinates $C^{-1}A_TC\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix} $ in $\alpha'$.

    So, to solve your problem:
    (1) write the transition matrix $C$ from $\alpha$ to $\alpha'$,
    (2) compute $C^{-1}A_TC$.
    Last edited by emakarov; May 5th 2014 at 11:38 AM. Reason: Replaced A with C.
    Thanks from Tweety
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