This preview shows page 1. Sign up to view the full content.
ASSIGNMENT 3  MATH235, FALL 2007
Submit by 16:00, Monday, October 1 (use the designated mailbox in Burnside Hall,
10
th
ﬂoor).
1.
The ring of
2
×
2
matrices over a ﬁeld
F
.
Let
F
be a ﬁeld. (In a ﬁrst read you may as well assume
F
is
R
, but solve the question in general). We
consider the set
M
2
(
F
) =
‰±
a b
c d
¶
:
a,b,c,d
∈
F
²
.
It is called the twobytwo matrices over
F
. We deﬁne the addition of two matrices as
±
a
1
b
1
c
1
d
1
¶
+
±
a
2
b
2
c
2
d
2
¶
=
±
a
1
+
a
2
b
1
+
b
2
c
1
+
c
2
d
1
+
d
2
¶
.
We deﬁne multiplication by
±
a
1
b
1
c
1
d
1
¶±
a
2
b
2
c
2
d
2
¶
=
±
a
1
a
2
+
b
1
c
2
a
1
b
2
+
b
1
d
2
c
1
a
2
+
d
1
c
2
c
1
b
2
+
d
1
d
2
¶
.
Prove that this is a ring. For each of the following subsets of
M
2
(
F
) determine if they are subrings or not.
h
A
i
The set
‰±
a b
0
d
¶
∈
M
2
(
F
)
²
.
h
B
i
The set
‰±
0
b
0 0
¶
∈
M
2
(
F
)
²
.
h
C
i
The set
‰±
a
0
c d
¶
∈
M
2
(
F
)
²
.
h
D
i
The set
‰±
a
0
0
d
¶
∈
M
2
(
F
)
²
.
h
E
i
The set
‰±
a
0
0
a
¶
∈
M
2
(
F
)
²
.
Remark: One deﬁnes in a very similar way the ring of
n
×
n
matrices with entries in a ﬁeld
F
.
2. Find the quotient and remainder when
a
is divided by
b
:
(1)
a
= 302
,b
= 19.
(2)
a
=

302
,b
= 19.
(3)
a
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 04/18/2008 for the course MATH 235 taught by Professor Goren during the Fall '07 term at McGill.
 Fall '07
 Goren
 Algebra, Matrices

Click to edit the document details