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Math Help - Algebra in the Einstein notation

  1. #1
    Junior Member RaisinBread's Avatar
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    Algebra in the Einstein notation

    Greetings,

    I'm having a little difficulty doing some basic exercises in the Einstein notation. The exercises in particular are attached to this post.

    In exercise 0, I get that, for example, $A^i_j$ refers to the entry located in row i column j of a change of basis matrix A. To solve the exercise, I first do;
    $\forall j, T(v_j)=T(A^i_j\tilde{v}_i)=A^i_jT(\tilde{v}_i)$, since T is linear. Now if I understand this correctly, I think;
    $A^i_jT(\tilde{v}_i)=A^1_jT(\tilde{v}_1)+...+A^n_j  T(\tilde{v}_n)$. Now I get that the next step I must do is write all of these $T(\tilde{v}_i)=\tilde{T}^i_k\tilde{v}_i$. However i'm having trouble writing the two nested sums with the Einstein notation. Intuitively I just feel like writing; $T(v_j)=A^i_j\tilde{T}^k_i\tilde{v_k}$, because for a fixed j, we sum over the i, and for every of these i, we sum over the k. From there However, I don't know what to do with $T(v_j)=T^i_jv_i=A^i_j\tilde{T}^k_i\tilde{v_k}$, my indices get very confusing and crazy, and I'm not sure what I can be allowed to simplify or all.

    Anyway, I hope all of this is understandable.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Algebra in the Einstein notation

    Quote Originally Posted by RaisinBread View Post
    Greetings,

    I'm having a little difficulty doing some basic exercises in the Einstein notation. The exercises in particular are attached to this post.

    In exercise 0, I get that, for example, $A^i_j$ refers to the entry located in row i column j of a change of basis matrix A. To solve the exercise, I first do;
    $\forall j, T(v_j)=T(A^i_j\tilde{v}_i)=A^i_jT(\tilde{v}_i)$, since T is linear. Now if I understand this correctly, I think;
    $A^i_jT(\tilde{v}_i)=A^1_jT(\tilde{v}_1)+...+A^n_j  T(\tilde{v}_n)$. Now I get that the next step I must do is write all of these $T(\tilde{v}_i)=\tilde{T}^i_k\tilde{v}_i$. However i'm having trouble writing the two nested sums with the Einstein notation. Intuitively I just feel like writing; $T(v_j)=A^i_j\tilde{T}^k_i\tilde{v_k}$, because for a fixed j, we sum over the i, and for every of these i, we sum over the k. From there However, I don't know what to do with $T(v_j)=T^i_jv_i=A^i_j\tilde{T}^k_i\tilde{v_k}$, my indices get very confusing and crazy, and I'm not sure what I can be allowed to simplify or all.

    Anyway, I hope all of this is understandable.
    Right...why are you using this notation, haha? Anyways, if I'm interpreting this correctly A is the change of basis matrix and so \widetilde{T}=ATA^{-1}, can you work out the indices for that?

    I'm not sure what one means per say. Is it saying what is the rule from changing from the basis \{f^i\} to \{\widetilde{f}^i\} in V^\ast?
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  3. #3
    Junior Member RaisinBread's Avatar
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    Re: Algebra in the Einstein notation

    I'm really not a fan of this notation, I'll admit, but this is a tensor analysis course, which apparently has a lot of applications in theoretical physics, so I suppose we are using notation that physicists use.

    As for problem 1, I think they mean that, suppose you have a vector g $\in$ V^*, then g can be expressed in both bases as; g=x_if^i=\tilde{x}_i\tilde{f}^i where the x_i are the coordinates of g in the \{f^i\} basis and the \tilde{x}_i are the coordinates in the \{\tilde{f}^i\} basis. I think that they want us to find a matrix B such that for every j, x_j=B^i_j\tilde{x_i}
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