Change of basis


This book [1] 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:


[1] Nadir Jeevanjee. An introduction to tensors and group theory for physicists. Springer, 2011.