<s>
In	O
linear	B-Language
algebra	I-Language
,	O
particularly	O
projective	O
geometry	O
,	O
a	O
semilinear	B-Algorithm
map	I-Algorithm
between	O
vector	O
spaces	O
V	O
and	O
W	O
over	O
a	O
field	O
K	O
is	O
a	O
function	O
that	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
"	O
up	O
to	O
a	O
twist	O
"	O
,	O
hence	O
semi-linear	O
,	O
where	O
"	O
twist	O
"	O
means	O
"	O
field	O
automorphism	O
of	O
K	O
"	O
.	O
</s>
<s>
)	O
,	O
it	O
may	O
be	O
termed	O
a	O
semilinear	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
The	O
special	O
case	O
where	O
the	O
field	O
is	O
the	O
complex	O
numbers	O
and	O
the	O
automorphism	O
is	O
complex	O
conjugation	O
,	O
a	O
semilinear	B-Algorithm
map	I-Algorithm
is	O
called	O
an	O
antilinear	O
map	O
.	O
</s>
<s>
Similar	O
notation	O
(	O
replacing	O
Latin	O
characters	O
with	O
Greek	O
)	O
are	O
used	O
for	O
semilinear	O
analogs	O
of	O
more	O
restricted	O
linear	B-Architecture
transform	I-Architecture
;	O
formally	O
,	O
the	O
semidirect	O
product	O
of	O
a	O
linear	O
group	O
with	O
the	O
Galois	O
group	O
of	O
field	O
automorphism	O
.	O
</s>
<s>
Indeed	O
every	O
linear	B-Architecture
map	I-Architecture
can	O
be	O
converted	O
into	O
a	O
semilinear	B-Algorithm
map	I-Algorithm
in	O
such	O
a	O
way	O
.	O
</s>
<s>
Thus	O
,	O
the	O
homothety	B-Algorithm
need	O
not	O
be	O
a	O
linear	B-Architecture
map	I-Architecture
,	O
but	O
is	O
-semilinear	O
.	O
</s>
<s>
Given	O
a	O
vector	O
space	O
V	O
,	O
the	O
set	O
of	O
all	O
invertible	O
semilinear	B-Algorithm
transformations	I-Algorithm
(	O
over	O
all	O
field	O
automorphisms	O
)	O
is	O
the	O
group	O
ΓL(V )	O
.	O
</s>
<s>
where	O
Aut(K )	O
is	O
the	O
automorphisms	O
of	O
K	O
.	O
Similarly	O
,	O
semilinear	O
transforms	O
of	O
other	O
linear	O
groups	O
can	O
be	O
defined	O
as	O
the	O
semidirect	O
product	O
with	O
the	O
automorphism	O
group	O
,	O
or	O
more	O
intrinsically	O
as	O
the	O
group	O
of	O
semilinear	B-Algorithm
maps	I-Algorithm
of	O
a	O
vector	O
space	O
preserving	O
some	O
properties	O
.	O
</s>
<s>
We	O
identify	O
Aut(K )	O
with	O
a	O
subgroup	O
of	O
ΓL(V )	O
by	O
fixing	O
a	O
basis	O
B	O
for	O
V	O
and	O
defining	O
the	O
semilinear	B-Algorithm
maps	I-Algorithm
:	O
</s>
<s>
Every	O
linear	B-Architecture
map	I-Architecture
is	O
semilinear	O
,	O
thus	O
.	O
</s>
<s>
The	O
projective	O
geometry	O
of	O
a	O
vector	O
space	O
V	O
,	O
denoted	O
PG(V )	O
,	O
is	O
the	O
lattice	O
of	O
all	O
subspaces	O
of	O
V	O
.	O
Although	O
the	O
typical	O
semilinear	B-Algorithm
map	I-Algorithm
is	O
not	O
a	O
linear	B-Architecture
map	I-Architecture
,	O
it	O
does	O
follow	O
that	O
every	O
semilinear	B-Algorithm
map	I-Algorithm
induces	O
an	O
order-preserving	O
map	O
.	O
</s>
<s>
That	O
is	O
,	O
every	O
semilinear	B-Algorithm
map	I-Algorithm
induces	O
a	O
projectivity	B-Algorithm
.	O
</s>
<s>
Thus	O
semilinear	B-Algorithm
maps	I-Algorithm
are	O
useful	O
because	O
they	O
define	O
the	O
automorphism	O
group	O
of	O
the	O
projective	O
geometry	O
of	O
a	O
vector	O
space	O
.	O
</s>
