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.