Homework Assignment #4
13.12 Write the following quadratic forms in matrix form:
a
)
x
2
1

2
x
1
x
2
+
x
2
2
.
b
) 5
x
2
1

10
x
1
x
2

x
2
2
.
c
)
x
2
1
+ 2
x
2
2
+ 3
x
2
3
+ 4
x
1
x
2

6
x
1
x
3
+ 8
x
2
x
3
.
Answer:
If we require the matrices to be symmetric, the solutions are:
±
1

1

1
1
²
,
±
5

5

5

1
²
,
1
2

3
2
2
4

3
4
3
.
13.21 Let
f
:
R
k
→
R
1
be continuous at the point
a
= (
a
1
,... ,a
k
). Consider the function
g
:
R
1
→
R
1
deﬁned
by
g
(
t
) =
f
(
t,a
2
,... ,a
k
). Show that
g
is continuous at
a
1
. This result implies that if
f
is continuous,
its restriction to any line parallel to a coordinate axis is also continuous. However, the converse is not
true. Consider the function
f
(
x,y
) =
xy
2
/
(
x
2
+
y
4
). Show that
f
1
(
t
) =
f
(
t,a
) and
f
2
(
t
) =
f
(
a,t
) are
continuous functions of
t
for each ﬁxed
a
. Show that
f
itself is not continuous at (0
,
0). [Hint: Take a
sequence on the diagonal.]
Answer:
Let
t
n
→
t
. Then (
t
n
,a
2
,... ,a
k
)
→
(
t,a
2
,... ,a
k
). Since
f
is continuous,
g
(
t
n
) =
f
(
t
n
,a
2
,... ,a
k
)
→
f
(
t,a
2
,... ,a
k
) =
g