Chapter 2
An Introduction to Vector
Spaces
1. Suppose
{
v
1
, . . . , v
k
}
is a linearly dependent set. Then show that one
of the vectors must be a linear combination of the others.
Answer 2.1
Since
{
v
1
, . . . , v
k
}
is a linearly dependent set of vectors,
there exist
k
scalars
α
1
, . . . , α
k
, not all zero, such that
α
1
v
1
+
···
+
α
k
v
k
=0
. By reindexing the vectors and scalars if necessary, we can
ensure that
α
1
±
. Then
v
1
=

1
α
1
(
α
2
v
2
+
+
α
k
v
k
)=
γ
2
v
2
+
+
γ
k
v
k
, where
γ
i
=

α
i
α
1
for
i
∈ {
2
, . . . , k
}
.
2. Let
x
1
,x
2
, . . . , x
k
∈
IR
n
be nonzero mutually orthogonal vectors. Show
that
{
x
1
, . . . , x
k
}
must be a linearly independent set.
Answer 2.2
This problem is equivalent to showing that
X
T
X
is non
singular, where
X
=(
x
1
, . . . , x
k
)
∈
n
×
k
. But
X
T
X
x
1
, . . . , x
k
)
T
(
x
1
, . . . , x
k
(
x
T
i
x
j
)
i,j
∈
k
, which is a diagonal
k
×
k
matrix whose diagonal entries
x
T
i
x
i
are nonzero since each
x
i
is nonzero. Thus
X
T
X
is nonsingular.
3. Let
v
1
, . . . , v
n
be orthonormal vectors in IR
n
. Show that
Av
1
, . . . , Av
n
are also orthonormal if and only if
A
∈
n
×
n
is orthogonal.
Answer 2.3
DeFne
V
v
1
, . . . , v
n
)
∈
n
×
n
.
The matrix
V
is
square with orthonormal columns, so the rank of
V
is
n
. Thus
V
has an
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