Then
𝐻
′
=
x
0
+
𝑉
′
is an aﬃne set (Exercise 1.150) which strictly contains
𝐻
. This contradicts the
definition of
𝐻
as a maximal proper aﬃne set.
5. Let
x
1
/
∈
𝑉
. By the previous part,
x
∈
lin
{
x
1
, 𝑉
}
. That is, there exists
𝛼
∈ ℜ
such that
x
=
𝛼
x
1
+
v
for some
v
∈
𝑉
To see that
𝛼
is unique, suppose that there exists
𝛽
∈ ℜ
such that
x
=
𝛽
x
1
+
v
′
for some
v
′
∈
𝑉
Subtracting
0
= (
𝛼
−
𝛽
)
x
1
+ (
v
−
v
′
)
which implies that
𝛼
=
𝛽
since
x
1
/
∈
𝑉
.
1.154
Assume
x
,
y
∈
𝑋
. That is,
x
,
y
∈ ℜ
𝑛
and
∑
𝑖
∈
𝑁
𝑥
𝑖
=
∑
𝑖
∈
𝑁
𝑦
𝑖
=
𝑤
(
𝑁
)
For any
𝛼
∈ ℜ
,
𝛼
x
+ (1
−
𝛼
)
y
∈ ℜ
𝑛
and
∑
𝑖
∈
𝑁
𝛼𝑥
𝑖
+ (1
−
𝛼
)
𝑦
𝑖
=
𝛼
∑
𝑖
∈
𝑁
𝑥
𝑖
+ (1
−
𝛼
)
∑
𝑖
∈
𝑁
𝑦
𝑖
=
𝛼𝑤
(
𝑁
) + (1
−
𝛼
)
𝑤
(
𝑁
)
=
𝑤
(
𝑁
)
Hence
𝑋
is an aﬃne subset of
ℜ
𝑛
.
1.155
See Exercise 1.129.
1.156
No.
A straight line through any two points in
ℜ
𝑛
+
extends outside
ℜ
𝑛
+
.
Put
differently, the aﬃne hull of
ℜ
𝑛
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 Fall '10
 Dr.DuMond
 Macroeconomics, All rights reserved, X1, Michael Carter, r????, Foundations of Mathematical Economics

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