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Math 235
Assignment 9 Solutions
1.
Let
~u
= (

2

3
i,
2 +
i
) and
~v
= (4

i,
4 +
i
). Use the standard inner product on
C
n
to
calculate
< ~u,~v >
,
< ~v,~u >
, and
k
~v
k
.
Solution: We have
< ~u,~v >
= (

2

3
i,
2 +
i
)
·
(4

i,
4 +
i
) = (

2

3
i
)(4 +
i
) + (2 +
i
)(4

i
)
=

5

14
i
+ 9 + 2
i
= 4

12
i
< ~v,~u >
= (4

i,
4 +
i
)
·
(

2

3
i,
2 +
i
) = (4

i
)(

2 + 3
i
) + (4 +
i
)(2

i
)
=

5 + 14
i
+ 9

2
i
= 4 + 12
i
k
~v
k
=
p
< ~v,~v >
=
q
(4

i,
4 +
i
)
·
(4

i,
4 +
i
)
=
p
(4

i
)(4 +
i
) + (4 +
i
)(4

i
) =
√
34
2.
Determine which of the following matrices is unitary.
a)
A
=
±
(1 +
i
)
/
√
7

5
/
√
35
(1 + 2
i
)
/
√
7 (3 +
i
)
/
√
35
²
.
Solution: Let
~u
= ((1 +
i
)
/
√
7
,
(1 + 2
i
)
/
√
7), and
~v
= (

5
/
√
35
,
(3 +
i
)
/
√
35). Then we
have
< ~u,~u >
=
1
7
[(1 +
i
)(1

i
) + (1 + 2
i
)(1

2
i
)] =
1
7
[2 + 5] = 1
< ~v,~v >
=
1
35
[(

5)(

5) + (3 +
i
)(3

i
)] =
1
35
[25 + 10] = 1
< ~u,~v >
=
1
√
7
√
35
[(1 +
i
)(

5) + (1 + 2
i
)(3

i
)] =
1
√
7
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This note was uploaded on 10/12/2010 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.
 Fall '08
 CELMIN
 Math

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