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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, $\displaystyle $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;

$\displaystyle $\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;

$\displaystyle $A^i_jT(\tilde{v}_i)=A^1_jT(\tilde{v}_1)+...+A^n_j T(\tilde{v}_n)$$. Now I get that the next step I must do is write all of these $\displaystyle $T(\tilde{v}_i)=\tilde{T}^i_k\tilde{v}_i$$. However i'm having trouble writing the two nested sums with the Einstein notation. Intuitively I just feel like writing;$\displaystyle $T(v_j)=A^i_j\tilde{T}^k_i\tilde{v_k}$$, because for a fixed j, we sum over the i, and for every of these i, we sum over the k. From there However, I don't know what to do with $\displaystyle $T(v_j)=T^i_jv_i=A^i_j\tilde{T}^k_i\tilde{v_k}$$, 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.

Re: Algebra in the Einstein notation

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Originally Posted by

**RaisinBread** 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, $\displaystyle $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;

$\displaystyle $\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;

$\displaystyle $A^i_jT(\tilde{v}_i)=A^1_jT(\tilde{v}_1)+...+A^n_j T(\tilde{v}_n)$$. Now I get that the next step I must do is write all of these $\displaystyle $T(\tilde{v}_i)=\tilde{T}^i_k\tilde{v}_i$$. However i'm having trouble writing the two nested sums with the Einstein notation. Intuitively I just feel like writing;$\displaystyle $T(v_j)=A^i_j\tilde{T}^k_i\tilde{v_k}$$, because for a fixed j, we sum over the i, and for every of these i, we sum over the k. From there However, I don't know what to do with $\displaystyle $T(v_j)=T^i_jv_i=A^i_j\tilde{T}^k_i\tilde{v_k}$$, 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 $\displaystyle A$ is the change of basis matrix and so $\displaystyle \widetilde{T}=ATA^{-1}$, 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 $\displaystyle \{f^i\}$ to $\displaystyle \{\widetilde{f}^i\}$ in $\displaystyle V^\ast$?

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