2.1.1. Basic formalism For all coordinate types, the first step
is a linear transformation applied via matrix multiplication of
the vector of pixel coordinate elements, pj: where rj are the
pixel coordinate elements of the reference point given by the
CRPIXj. Henceforth we will consistently use j for pixel axis
indexing and i for the world axes. The mij matrix is a non-
singular square matrix of dimension N × N. In the first
instance, N is given by the NAXIS keyword value, but this will
be generalized with the introduction of the WCSAXES keyword in
Sect. 2.2 so that the dimension of the world coordinates need
not match that of the data array. The elements, qi, of the
resulting intermediate pixel coordinate vector are offsets, in
dimensionless pixel units, from the reference point along axes
coincident with those of the intermediate world coordinates.
Thus the conversion of qi to the corresponding intermediate
world coordinate element, xi, is a simple scale: We defer
discussion of the encoding of mij and si as