Math 113 Homework # 3, selected solutions
Fraleigh 5.13: Yes, this set of matrices, which is called O(
n
), is a subgroup
of GL(
n,
R
). First note that O(
n
) is a sub
set
of GL(
n,
R
) because if
A
T
A
=
I
, then
A
is injective, so by the “ranknullity” theorem in linear
algebra, its range has dimension
n
, so
A
is invertible. We also note that
since left inverses are unique,
A
T
=
A

1
, so
AA
T
=
I
also. Now O(
n
)
is closed under multiplication because if
A
T
A
=
I
and
B
T
B
=
I
then
(
AB
)
T
(
AB
) =
B
T
A
T
AB
=
B
T
IB
=
B
T
B
=
I
.
O(
n
) contains the
identity because
I
T
I
=
I
2
=
I
. O(
n
) is closed under inverses because
if
A
T
A
=
I
then (
A

1
)
T
A

1
= (
A
T
)

1
A

1
= (
AA
T
)

1
=
I

1
=
I
.
(Here we have used the facts that
AA
T
=
I
as explained above, and
also that for any invertible matrix
A
we have (
A
T
)

1
= (
A

1
)
T
; this
holds because (
A

1
)
T
A
T
= (
AA

1
)
T
=
I
T
=
I
.)
Fraleigh 5.54:
H
∩
K
is closed under multiplication because if
x, y
∈
H
∩
K
,
then since
x, y
∈
H
and
H
is a subgroup we have
xy
∈
H
; since
x, y
∈
K
and
K
is a subgroup we have
xy
∈
K
; so
xy
∈
H
∩
K
.
H
∩
K
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 Fall '08
 OGUS
 Math, Matrices, Zn, Cyclic group, Fi Fj

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