One of the simplest linear transformations is a domain coordinate transformation, which are usually implemented by the anatomical pattern of projections from one area to another. These operations transform the coordinates of the field's domain, thus distorting the shape of the field, perhaps for some information processing end or for a more efficient allocation of ``neural real estate.'' (An example, the ``logmap transformation'' in the primary visual cortex, is discussed below.)
In general, if 
is a mapping from coordinates in
region A to coordinates in region B, then the activity
field 
defined over B, which is induced by activity
field 
over A, is given by 
,
that is, for any coordinates 
, 
.
Thus, if we ignore scaling of amplitudes, the activity
induced by the projection at h(p) in B is equal to the
source activity at p in A.
Most such coordinate transformations are ``one-to-one and
onto,'' in which cases we can define the induced activity
field directly:
, or
[ (q) = [h^-1(q)] ]
for all 
.
That is, the activity at q in B is given by the activity
at 
in A.
(Note that the field transformation from 
to 
is
linear even if the coordinate transformation h is not.)
For example, a coordinate transformation, the logmap
transformation (Baron 1987, pp. 181-186), takes place
between the retina and its first projection in the primary
visual cortex (VI).
If retinal coordinates are represented by a complex number
z in polar coordinates (giving an angle and distance from
the center of the retina), then the field 
in VI is
related to the retinal field 
by
[ (z) = ( e^z ) ,]
where 
is the complex exponential function.
The effect of this is
,
that is, radial distance is transformed logarithmically.
In addition to devoting more ``neural real estate'' to the center
of the retina, this transformation has the effect of
converting rotations and scale changes of centered images
into simple
translations (Schwartz 1977, Baron 1987, ch. 8).
To see this, note that if 
is a scaled
version of 
, then the corresponding VI field is
[ '(z) = '(z) = (sz) = (sz)
= [(s) + (z)] ,]
which is 
, the image of 
, translated by
.
Similarly, if 
is a rotation
of 
through angle 
, then the corresponding
field is
[ '(z) = '(z) = (e^i 
z)
= [(e^i 
z)]
= [i 
+ z] ,]
which is 
, the image of 
, translated by
(in a perpendicular direction to the other
translation).