Homework 3 Solutions
1. Prove that the set of all collineations in
Trans
(
R
n
) is a subgroup of
Trans
(
R
n
).
Solution:
We need to show that the collineations satisfy the identity, inverse, and clo
sure properties (under function composition). It is immediate from the deﬁnition that
ι
is a collineation. In Problem 2 of Homework 2 it was proved that if
α
∈
Trans
(
R
n
) is a
collineation, then
α

1
is a collineation, so the inverse property is satisﬁed. Finally, if
α,β
are collinations, then let
‘
be a line and set
‘
0
=
β
(
‘
) and
‘
00
=
α
(
‘
0
). Now
‘
0
is a line
since
β
is a collineation and then
‘
00
is a line since
α
is a collineation, thus we ﬁnd that
α
◦
β
(
‘
) =
α
(
β
(
‘
)) =
α
(
‘
0
) =
‘
00
. Thus, the image of
‘
under
αβ
is a line and since
‘
was
arbitrary, we ﬁnd that
αβ
is a collineation.
2. Find an involution in
Trans
(
R
2
) which is not the identity, a reﬂection, or a rotation.
Solution:
The function
f
:
R
2
→
R
2
given by
f
(
x,y
) =
(
x,y
) if (
x,y
)
6
= (0
,
0)
,
(1
,
1)
(1
,
1) if (
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 Spring '11
 RWK
 Rotation, Invertible matrix

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