A *projective basis* for
is any set of *n*+2 points of
,
no *n*+1 of which lie in a hyperplane. Equivalently, the
matrix formed by the column vectors of any *n*+1 of
the points must have full rank.

It is easily checked that
forms a basis, called the *canonical basis*. It contains the
points at infinity along each of the *n* coordinate axes, the origin,
and the *unit point*
.
Any basis can be
mapped into this standard form by a suitable collineation.

A full proof can be found in [23]. We will just check
that there are the right number of constraints to uniquely characterize the
collineation. This is described by an
matrix
*A*, defined up to an overall scale factor, so it has
(*n* + 1)^{2}-1=
*n*(*n*+2) degrees of freedom. Each of the *n*+2 basis point images
provides *n* constraints (*n*+1 linear equations
defined up a common scale factor), so the required total of *n*(*n*+2)constraints is met.

An analogous relation holds for the unit point in the canonical basis.