This preview shows page 1. Sign up to view the full content.
EE221A Problem Set 1 Solutions  Fall 2011
Note: these solutions are somewhat more terse than what we expect you to turn in, though the important thing is
that you communicate the main idea of the solution.
Problem 1. Functions.
It is a function; matrix multiplication is well deﬁned. Not injective; easy to ﬁnd
a counterexample where
f
(
x
1
) =
f
(
x
2
)
;
x
1
=
x
2
. Not surjective; suppose
x
= (
x
1
,x
2
,x
3
)
T
.
Then
f
(
x
) =
(
x
1
+
x
3
,
0
,x
2
+
x
3
)
T
; the range of
f
is not the whole codomain.
Problem 2. Fields.
a) Suppose
0
0
and
0
are both additive identities. Then
x
+ 0
0
=
x
+ 0 = 0
⇐⇒
0
0
= 0
.
Suppose
1
and
1
0
are both multiplicative identities. Consider for
x
6
= 0
,
x
·
1 =
x
=
x
·
1
0
. Premultiply by
x

1
to
see that
1 = 1
0
.
b) We are not given what the operations
+
and
·
are but we can assume at least that
+
is componentwise addition.
The identity matrix
I
is nonsingular so
I
∈
GL
n
. But
I
+ (

I
) = 0
is singular so it cannot be a ﬁeld.
Problem 3. Vector Spaces.
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '10
 ClaireTomlin

Click to edit the document details