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,

refers to the entry located in row i column j of a change of basis matrix A. To solve the exercise, I first do;

, since T is linear. Now if I understand this correctly, I think;

. Now I get that the next step I must do is write all of these

. However i'm having trouble writing the two nested sums with the Einstein notation. Intuitively I just feel like writing;

, 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

, 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.