Solutions to PS 9 (Math 121)
J. Scholtz
January 6, 2007
Question 1: 6.1/20
Prove the Polar Identities:
1.

x
+
y

2
+

x

y

2
=
h
x
+
y,x
+
y
ih
x

y,x

y
i
=
h
x,x
i
+
h
y,y
i
+2
h
x,y
i
+2
h
y,x
ih
x,x
ih
y,y
i
=
4
h
x,y
i
.
2.

x
+
y

2
+
i

x
+
iy

2

x

y

2

i

x

iy

2
= (

x
+
y

2

x

y

2
)+
i
(

x
+
iy

2

x

iy

2
) = 4
h
x,y
i
.
Question 2: 6.1/24d
Show that

(
a,b
)

= max
{
a

,

b
}
is a norm.
1. Positivedeﬁnitness: clearly by deﬁnition it is positive. Also the only way max
{
a

,

b
}
= 0 is when
(
a,b
) = (0
,
0).
2.

c
(
a,b
)

=

(
ca,cb
)

= max
{
ca

,

cb
}
=

c

max
{
a

,

b
}
=

c

(
a,b
)

.
3. Note that

a
+
c

<

a

+

c

and

b
+
d

<

b

+

d

and so the inequality holds.
Question 3: 6.1/25
If the inner product was deﬁned using the polar identity and the deﬁnition of the norm from 24(d) then
it would not be linear. It would be suﬃcient to show this on the pairs
x
= (1
,
0) and
y
= (2
,
2):
h
(1
,
0)
,
(2
,
2)
i
=
1
4
(9

4) =
5
4
6
= 2
h
(1
,
0)
,
(1
,
1)
i
=
1
2
(4

1) =
3
2
Question 4: 6.2/2(i)
I am not including the solution since it is just computation. However, if you ﬁnd this any problems with
this question do not hesitate to contact me.
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 Spring '08
 GUREVITCH
 Math, Linear Algebra, Vector Space, Dot Product, Hilbert space

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