AMS210.01.
Homework 6
Andrei Antonenko
Due at the beginning of the class, April 28, 2003
In this homework we will consider the following scalar products
•
In
R
n
:
h
(
x
1
,x
2
,...,x
n
)
,
(
y
1
,y
2
,...,y
n
)
i
=
x
1
y
1
+
x
2
y
2
+
···
+
x
n
y
n
;
•
In
M
m,n
:
h
A,B
i
= tr(
AB
>
);
•
In
C
[
a,b
]:
h
f,g
i
=
R
b
a
f
(
t
)
g
(
t
)
dt
.
In this homework there are a lot of extracredit problems of diﬀerent levels of diﬃculty. You should
understand perfectly how to solve standard problems, and proceed to extracredit problems only after
that! The problems from exam and quizes will include only standard problems.
1. Compute
h
3
u

5
v,
2
u
+
v
i
if
h
u,u
i
= 5
,
h
u,v
i
= 1 and
h
v,v
i
= 2.
2. Compute the following scalar products:
(a)
h
(2
,
1
,

3
,
1)
,
(0
,
4
,
2
,
2)
i
in
R
4
with standard scalar product.
(b)
h
2
t
+ 1
,t
2
i
in the space
C
[0
,
1].
(c)
h
2
t
+ 1
,t
2
i
in the space
C
[

1
,
1].
3. Find norms and normalizations of the following vectors:
(a) (4
,
2
,
2) in
R
3
with standard scalar product.
(b)
t
2
in
C
[0
,
1].
(c)
t
2
in
C
[

1
,
1].
4. Find the cosines of the angles between the following vectors:
(a) (0
,
2) and (3
,

3).
(b)
t
2
and
t
2
+ 1 in
C
[0
,
1].
(c)
ˆ
1 2 3
4 5 6
!
and
ˆ
4 5 6
1 2 3
!
in
M
2
,
3
.
5. Find all constants
a
such that
(a) vectors (
a,
2) and (
a,

8) are orthogonal in
R
2
.
1