Math 104
Fall 2009
Homework 3: solution set
We use the notations below to ease readability.
Matrices are bold capital, vectors are bold lowercase
and scalars or entries are not bold.
For instance,
A
is a matrix and
a
ij
(sometimes,
A
(
i, j
)) its (
i, j
)th
entry. Likewise
x
is a vector and
x
j
its
j
th component. The linear span generated by a group of vectors
v
1
,
v
2
, . . . ,
v
n
is denoted by span(
v
1
,
v
2
, ...,
v
n
).
Problem 1
null space; left null space.
Problem 2
Set
v
1
=
1
1
1
and
v
2
=
1
0
1
.
We proceed by finding an orthonomal basis of span(
v
1
,
v
2
).
Let
u
1
=
v
1

v
2
=
0
1
0
and
u
2
=
v
2
k
v
2
k
=
1
√
2
1
0
1
. Then
u
1
,
u
2
∈
span(
v
1
,
v
2
),
k
u
1
k
=
k
u
2
k
= 1 and
u
*
1
u
2
= 0.
Since dim(span(
v
1
,
v
2
)) = 2,
{
u
1
,
u
2
}
is an orthonormal basis of span(
v
1
,
v
2
). The orthogonal projection
onto span(
u
1
,
u
2
) is
P
=
u
1
u
*
1
+
u
2
u
*
2
=
1
2
0
1
2
0
1
0
1
2
0
1
2
.
Problem 3
(a) Suppose
λ
is an eigenvalue of
A
. Then there is a nonzero vector
x
∈
C
m
obeying
Ax
=
λ
x
, and
x
*
Ax
=
x
*
λ
x
=
λ
k
x
k
2
.
(1)
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 Fall '09
 Linear Algebra, Vectors, Matrices, Scalar, Hilbert space, αuv ∗

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