EE221A Section 2
9/2/11
1
Fields
1. Show that the set
{
0
,
1
}
, with multiplication deﬁned as binary AND and addition
deﬁned as binary XOR, is a ﬁeld.
×
(AND)
0
1
0
0
0
1
0
1
+
(XOR)
0
1
0
0
1
1
1
0
2. Show that
∀
α
∈
F
,
α
·
0 = 0
·
α
= 0
.
2
Vector Spaces
1. Does
C
form a vector space over
R
? Does
R
form a vector space over
C
?
2. Show that a set
W
⊂
V
is a subspace of
V
iﬀ
∀
α,β
∈
F
,
∀
x,y
∈
W
,
αx
+
βy
∈
W
.
3. Show that the additive identity
θ
V
of a vector space
V
is unique.
4. Consider
V
= (
C
([0
,
1]
,
R
)
,
R
)
, the vector space of continuous, realvalued func
tions on the interval
[0
,
1]
, over the reals. Prove that
D
:=
±
f
(
·
)
∈
V
:
ˆ
1
0
f
(
τ
)
dτ
= 0
²
is a subspace of
V
.
1
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View Full Document3 Linear independence
2
3
Linear independence
1. Show that the set of polynomials
{
2
x

1
,x
2
+ 2
x

2
,
3
}
is linearly independent
in
R
[
x
]
, the set of polynomials with coeﬃcients in
R
.
2. Show that
{
e
t
,e
2
t
}
is linearly independent in
V
= (
C
([0
,
1]
,
R
)
,
R
)
.
4
Basis and Coordinate Representation
[last 3 pages of Lecture Notes 2]
•
Basis:
B
=
{
b
1
,b
2
,...,b
n
}
is a
basis
of
V
if
–
(i)
B
spans
V
;
–
(ii)
B
is linearly independent
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 Fall '10
 ClaireTomlin
 Linear Algebra, Vector Space, coordinate representation

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