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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, 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.

Re: Algebra in the Einstein notation

Quote:

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,

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.

Right...why are you using this notation, haha? Anyways, if I'm interpreting this correctly is the change of basis matrix and so , 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 to in ?

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 , then g can be expressed in both bases as; where the are the coordinates of g in the basis and the are the coordinates in the basis. I think that they want us to find a matrix such that for every j,