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Theorem 34
(Thm 13.3, p.291)
.
The general quadratic form Q
(
x
1
,
···
,
x
k
) =
∑
i
≤
j
a
i j
x
i
x
j
can be written as
(
x
1
x
2
···
x
k
)
a
11
1
2
a
12
···
1
2
a
1
k
1
2
a
12
a
22
···
1
2
a
2
k
.
.
.
.
.
.
.
.
.
.
.
.
1
2
a
1
k
1
2
a
2
k
···
a
kk
x
1
x
2
.
.
.
x
k
,
or
x
T
A
x
, where A is a symmetric matrix.
11.2 Continuous Functions
Defnition 24
(p.293)
.
A function f
:
R
k
→
R
m
is
continuous at
x
0
if whenever
{
x
n
}
∞
n
=
1
converges to
x
0
the sequence
{
f
(
x
n
)
}
∞
n
=
1
in
R
m
converges to f
(
x
0
)
. The function f is
said to be
continuous
if it is continuous at every point in its domain.
Theorem 35
(Thm 13.5, p.295)
.
Let f
= (
f
1
,
···
,
f
m
)
be a function from
R
k
to
R
m
.
Then, f is continuous at
x
if and only if each of its component functions f
i
:
R
k
→
R
1
is continuous at
x
.
If
f
and
g
are two functions from
R
k
to
R
m
,
f
+
g
,
f

g
,
f
·
g
are deFned by
(
f
+
g
)(
x
) =
f
(
x
)+
g
(
x
)
,
(
f

g
)(
x
) =
f
(
x
)

g
(
x
)
,
(
f
·
g
)(
x
) =
f
(
x
)
·
g
(
x
)
,
respectively.
Theorem 36
(Thm 13.4, p.294)
.
Let f and g be functions from
R
k
to
R
m
. If f and g
are continuous at
x
, then f
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This note was uploaded on 03/26/2012 for the course ECON 205 taught by Professor Mr.lee during the Spring '11 term at Korea University.
 Spring '11
 Mr.Lee

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