Math 235
Assignment 10 Solutions
1.
Consider
C
3
with its standard inner product. Let
~
z
=
1 +
i
2

i

1 +
i
,
~w
=
1

i

2

3
i

1
.
a) Evaluate
h
~
z, ~w
i
and
h
~w,
2
i~
z
i
.
Solution: We have
h
~
z, ~w
i
= (1 +
i
)(1 +
i
) + (2

i
)(

2 + 3
i
) + (

1 +
i
)(

1) = 2
i

1 + 8
i
+ 1

i
= 9
i
h
~w,
2
i~
z
i
=
2
i
h
~
z, ~w
i
=

2
i
(

9
i
) =

18
b) Find a vector in span
{
~
z, ~w
}
that is orthogonal to
~
z
.
Solution: A vector orthogonal to
~
z
in span
{
~
z, ~w
}
is
~v
= perp
~
z
~w
=
~w

h
~w,~
z
i
k
~
z
k
2
~
z
=
1

i

2

3
i

1
+
9
i
9
1 +
i
2

i

1 +
i
=
0
1

i

2

i
c) Write the formula for the projection of
~u
onto
S
= span
{
~
z, ~w
}
.
Solution: Note that span
{
~
z, ~w
}
= span
{
~
z,~v
}
where
~v
is the vector from part b) and
{
~
z,~v
}
is orthogonal over
C
so we have
proj
S
~u
=
< ~u,~
z >
k
~
z
k
2
~
z
+
< ~u,~v >
k
~v
k
2
~v
2.
Let
V
be an inner product space, with complex inner product
h
,
i
. Prove that if
< ~u,~v >
= 0, then
k
~u
+
~v
k
2
=
k
~u
k
2
+
k
~v
k
2
. Is the converse true?
Solution: If
< ~u,~v >
= 0, then we have
k
~u
+
~v
k
2
=
< ~u
+
~v,~u
+
~v >
=
< ~u,~u
+
~v >
+
< ~v,~u
+
~v >
=
< ~u,~u >
+
< ~u,~v >
+
< ~v,~u >
+
< ~v,~v >
=
k
~u
k
2
+ 0 +
0 +
k
~v
k
2
=
k
~u
k
2
+
k
~v
k
2
The converse is not true. One counter example is: Consider
V
=
C
with its standard inner
product and let
~u
= 1 +
i
and
~v
= 1

i
. Then
k
~u
+
~v
k
2
=
k
2
k
2
= 4 and
k
~u
k
2
+
k
~v
k
2
=
2 + 2 = 4, but
< ~u,~v >
= (1 +
i
)(1 +
i
) = 2
i
6
= 0.
1
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3.
Prove that for any
n
×
n
matrix
A
, we have det
A
=
det
A
.
Solution: We prove this by induction. If
A
is a 1
×
1 matrix, then the result is obvious.
Assume the result holds for
n

1
×
n

1 matrices and consider an
n
×
n
matrix
A
. If we
expand det
A
along the ﬁrst row, we get by deﬁnition of the determinant
det
A
=
n
X
i
=1
a
1
i
C
1
i
(
A
)
.
where
C
1
i
(
A
) represents the cofactors of
A
. But, each of these cofactors is the determinant
of an
n

1
×
n

1 matrix, so we have by our inductive hypothesis that
C
1
i
(
A
) =
C
1
i
(
A
).
Hence,
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 Fall '08
 CELMIN
 Math, Linear Algebra, Algebra, Matrices, CN, Orthogonal matrix, Normal matrix, standard inner product

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