CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b; Solutions to Homework Set 1
Due: January 14, 2008
1. We must show
W
is nonempty and closed. By Remark 2 in class, 0
∈
U
for each
U
∈ U
, so 0
∈
W
.
Thus
W
6
=
∅
.
Let
x, y
∈
W
and
a, b
∈
F
.
Then
x, y
∈
U
for all
U
∈ U
, so as
U
is a subspace,
ax
+
by
∈
U
. As this holds for all
U
∈ U
,
ax
+
by
∈
W
, so
indeed
W
is closed.
2. The system
F
is
homogeneous
if
b
i
= 0 for all
i
. Claim
S
=
S
(
F
) is a subspace
iff
F
is homogeneous. First if
S
is a subspace then by Remark 2 in Chapter 1, 0
∈
S
.
But 0 = (0
, . . . ,
0), so
b
i
=
∑
j
a
i,j
·
0 =
∑
j
0 = 0 as
a
·
0 = 0 for all
a
∈
F
by 1.3 and
0 + 0 = 0 by definition of 0. Thus if
S
is a subspace then
F
is homogeneous.
Conversely assume
F
is homogeneous. Let
S
i
be the set of solutions to the
i
th equa
tion. We will show each
S
i
is a subspace of
V
n
. Then as
S
=
T
i
S
i
,
S
is a subspace by
Problem 1.
Let
u
= (
u
1
, . . . , u
n
) and
v
= (
v
1
, . . . , v
n
) be solutions to
a
1
x
1
+
· · ·
+
a
n
x
n
= 0 and
b, c
∈
F
. Then
a
1
(
bu
1
+
cv
1
) +
· · ·
+
a
n
(
bu
n
+
cv
n
) =
b
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 Winter '07
 Aschbacher
 Math, Linear Algebra, Algebra, Equations, Quadratic equation, Elementary algebra

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