Similarly to what has been done for vectors in section X.B.2.1, one derives transformations of the components of tensors. One considers the components of tensor in two coordinate systems:
Using the same definition of transformation matrix as in section X.B.2.1, one writes:
Then, the substitution of the two expressions in the equations defining components gives:
The last expression allows us to extract a relation involving only the components of tensors, and not the base vectors:
Here again, one recognizes a classical matricial expression: