A change of coordinates matrix, also called a transition matrix, specifies the transformation from one vector basis to another under a change of basis. For example, if and are two vector bases in , and let be the coordinates of a vector in basis and its coordinates in basis .
Write the basis vectors and for in coordinates relative to basis as
(1)
 
(2)

This means that
(3)
 
(4)

giving the change of coordinates matrix
(5)

which specifies the change of coordinates of a vector under the change of basis from to via
(6)
