**2.1.3. Usage comments**

The proposal presented in this and the subsequent papers is not simple
and provides wide latitude for mistakes in describing the WCS and in
writing the FITS headers. The result of an improperly described WCS
is simply undefined; it is the job of the FITS writer to produce a
correct description. A simple error which could be made in a WCS
description, or with other parts of a header, is a repetition of
keywords with different values assigned to them. If, for example,
`BUNIT` were repeated with a new value, the data would have
unknown units but would be read correctly. In binary tables, a second
value for `TFORM n` would cause the tabular data to be read
incorrectly.

This is a very general proposal! The linear transformation matrix
allows for skew and fully general rotations. The reader should note
that this allows dissimilar axes to be rotated into one another. This
is meaningful in imaging; for example, one may wish to re-sample a
spectral-line cube from some special viewing angle in the three-space
of two celestial coordinates and one frequency coordinate. Such
rotations are, however, forbidden into axes whose coordinate values
are, by convention, only integral. Thus, if
`CTYPE i`

The linear transformation matrix could also be used to represent images that have been transposed, e.g.

This is a legal usage, but likely to confuse the reader. In this
example, the FITS user will read in the header that the first element
of the world coordinate is `CTYPE1`, although this corresponds to
the second pixel axis. Note that keywords `NAXIS1`,
`CRPIX1`, `PC i_`, and

Equations (1) and (2) allow considerable
flexibility in the way the linear transformation is partitioned
between the `PC i_j` and

(4) |

for all *i*. This normalization leaves orthogonal matrices unchanged,
and only slightly modifies matrices which are nearly orthogonal. Note
that this is not the same as setting the determinant of the
`PC i_j` matrix to unity. Note also that this constraint
is optional and may not be the most physically meaningful selection of
the