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Math Help - Compute the change of basis matrix

  1. #1
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    Compute the change of basis matrix

    I wish to compute the change of basis for the following:

    B = {  \begin{bmatrix} 1\\ 1 \end{bmatrix},\begin{bmatrix} 1\\ -1 \end{bmatrix} }

    C = {  \begin{bmatrix} 0\\ 1 \end{bmatrix},\begin{bmatrix} 2\\ 1 \end{bmatrix} }

    How can I compute the change of basis matrix P_{C \leftarrow B}?

    There are supposedly (at least) two ways to do this.

    I know how to do this by expressing the vectors in B in terms of the basis C; when I do that I get the result that:

    P_{C \leftarrow B} = \begin{bmatrix} 1/2 & -3/2\\ 1/2 & 1/2 \end{bmatrix}

    Is this correct at all? Although even if it is correct I'd like to be able to do it using a different approach I got described.

    Supposedly I first have to transform the B coordinates to the canonical coordinates and thereafter the canonical coordinates to C coordinates, this should give P_{C \leftarrow B} as P_C^{-1}P_B
    Could someone show me how to do this for this example?
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  2. #2
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    Quote Originally Posted by posix_memalign View Post
    I wish to compute the change of basis for the following:

    B = {  \begin{bmatrix} 1\\ 1 \end{bmatrix},\begin{bmatrix} 1\\ -1 \end{bmatrix} }

    C = {  \begin{bmatrix} 0\\ 1 \end{bmatrix},\begin{bmatrix} 2\\ 1 \end{bmatrix} }

    How can I compute the change of basis matrix P_{C \leftarrow B}?

    There are supposedly (at least) two ways to do this.

    I know how to do this by expressing the vectors in B in terms of the basis C; when I do that I get the result that:

    P_{C \leftarrow B} = \begin{bmatrix} 1/2 & -3/2\\ 1/2 & 1/2 \end{bmatrix}

    Is this correct at all? Although even if it is correct I'd like to be able to do it using a different approach I got described.


    It is correct and hopefully you understand what you did here: you express each element of the new

    basis as a lin. combination of the new basis and the matrix we're looking for is the transpose of the

    coefficient matrix obtained in the first step.

    This is, as far as I can tell, the easiest and quickest method.



    Supposedly I first have to transform the B coordinates to the canonical coordinates and thereafter the canonical coordinates to C coordinates, this should give P_{C \leftarrow B} as P_C^{-1}P_B
    Could someone show me how to do this for this example?

    As I can see this is exactly the same as before: to express the basis B in terms of the canonical

    coordinates is just to take the matrix B formed with the elements of B, and then...etc. Exactly the same!

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    As I can see this is exactly the same as before: to express the basis B in terms of the canonical

    coordinates is just to take the matrix B formed with the elements of B, and then...etc. Exactly the same!

    Tonio
    Thank you for your reply.

    I know they are the same but I don't see how the second approach yields the result, i.e. I don't know how to apply that approach. It would be great with an example or even just a name of the technique so that I could find an example.
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