General

The basis vectors do not have to be orthogonal, as in the example. In general, suppose I have a basis S, . Then any arbitrary vector can be written as

where are the coordinates of in basis S. More briefly,

Suppose I have another basis S', .Then the same vector can also be written as

or

The relationship between the two sets of coordinates is always

where P is a matrix that is called the transformation matrix from S to S'. (Although it really works the opposite way.)

Matrix P takes the form:

It contains the basis vectors of the S' system written in the S system. (That is why if I multiply with P, I get a vector in the S system.)

To get the transformation the other way, use the matrix P^-1.