3.238
Let
a
𝑗
,
𝑗
= 1
,
2
, . . . , 𝑚
denote the rows of
𝐴
. Each
a
𝑖
defines linear functional
𝑔
𝑗
(
𝑥
) =
a
𝑗
𝑥
on
ℜ
𝑛
, and
c
defines another linear functional
𝑓
(
𝑥
) =
c
𝑇
x
. Assume that
𝑓
(
𝑥
) =
c
𝑇
x
= 0 for every
x
∈
𝑆
where
𝑆
=
{
x
:
𝑔
𝑗
(
x
) =
a
𝑖
x
= 0
, 𝑗
= 1
,
2
, . . ., 𝑚
}
Then the system
𝐴𝑥
= 0
has no solution satisfying the constraint
c
𝑇
x
>
0. By Exercise 3.20, there exists scalars
𝑦
1
, 𝑦
2
, . . . , 𝑦
𝑚
such that
𝑓
(
x
)
=
𝑚
∑
𝑗
=1
𝑦
𝑗
𝑔
𝑗
(
x
)
or
c
=
𝑚
∑
𝑗
=1
𝑦
𝑗
𝑎
𝑗
=
𝐴
𝑇
y
That is
y
= (
𝑦
1
, 𝑦
2
, . . . , 𝑦
𝑚
) solves the related nonhomogeneous system
𝐴
𝑇
y
=
c
Conversely, assume that
𝐴
𝑇
y
=
c
for some
𝑦
∈ ℜ
𝑚
. Then
c
𝑇
x
=
𝑦𝐴𝑥
= 0
for all
𝑥
such that
𝐴𝑥
= 0 and therefore there is no solution satisfying the constraint
c
𝑇
x
= 1.
3.239
Let
𝑆
=
{
z
:
z
=
𝐴
x
,
x
∈ ℜ }
the image of
𝑆
.
𝑆
is a subspace. Assume that system I has no solution, that is
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 Fall '10
 Dr.DuMond
 Macroeconomics, Logic, Optimization, All rights reserved, Functional, Foundations of Mathematical Economics

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