This book  uses an interesting notation for change of basis calculations where the two bases are distinguished by priming the indices.
Suppose for some -dimensional vector space we have an original basis and a primed basis (of course, the choice of calling one basis the original the other primed is arbitrary). There are linear transformations from each basis to the other:
Note that the position of the prime matters: if we consider and as matrices, they would be two different matrices. In fact, they would be inverses. Using the fact that for :
How does the dual basis transform between and ? Since each is itself a covector, we can express it in terms of its components in the original dual basis:
Then we determine what the are by evaluating on the basis vectors:
Plugging this result back into Eq 2 we get . A similar calculation gives the opposite change of dual basis:
Comparing this to Eq. 1, we see that the dual basis changes in the inverse manner as the basis.
For a vector , how do the vector components change from the original to primed basis? Recall that we can use the 'th dual basis to obtain the 'th component of a vector:
and similarly . Expanding using Eq. 3:
So (after a similar calculation in the opposite direction) we see vector components change as:
Now consider a (1,1)-tensor with components in the original basis. Unlike the book we use the convention that a -tensor is a multilinear map:
How do the tensor components transform from the original to primed basis? We can simply evaluate on the primed basis and expand: