This matrix is referred to as the basis transformation matrix from B to C. It can be regarded as an automorphism over . Any vector v represented in B can be transformed to a representation in C as follows:
:

Under the transformation of basis, the superscript on the transformation matrix, M, and the subscript on the coordinate vector, v, are the same, and seemingly cancel, leaving the remaining subscript. While this may serve as a memory aid, it is important no such cancellation, or similar