Algebra in the Einstein notation

**RaisinBread** · September 14th 2011, 02:53 PM

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.

**Drexel28** · September 14th 2011, 05:02 PM

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, $A^i_j$ 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 $V^\ast$ ?

**RaisinBread** · September 14th 2011, 08:35 PM

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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