Understanding Change of Basis in N-Dimensional Space

[Grand]

Homework Statement

Could someone help me understand the following manipulations concerning change of babsis in an N-dimensional space:
$$|i'\right\rangle=R|i\right\rangle=\sum_{j=1}^NR_{ji}|j\right\rangle$$
multiply around by $$(R^{-1})_{ik}$$
$$(R^{-1})_{ik}|i'\right\rangle=\sum_{j=1}^NR_{ji}(R^{-1})_{ik}|j\right\rangle$$
and then they say that after a change of indices, we get:
$$|i\right\rangle=\sum_{j=1}^NR^{-1}_{ji}|j' \right\rangle$$
However, I can't understand how do we gat the summation symbol on the LHS and how exactly do we do this change of indices.

Any help would be greatly appreciated.

 

[LightHero]

Homework Equations The equation given in the problem statement is |i'\right\rangle=R|i\right\rangle=\sum_{j=1}^NR_{ji}|j\right\rangleThe Attempt at a Solution I'm not sure how to solve this problem. I think that the change of indices is done by replacing the |i'\rangle on the left-hand side with |i\rangle, since they are equal according to the equation given in the problem statement. Then, the summation symbol on the left-hand side is added by using the fact that R|i\rangle can be written as a sum of terms.