125
(ii)
If
PF
(
k
)
⊆
F
(
k
)
denotes the family of all polynomial functions
a
→
α
n
a
n
+ · · · +
α
1
a
+
α
0
, prove that
PF
(
k
)
is a subspace of
F
(
k
)
.
Solution.
P
(
k
)
is closed under addition and scalar multiplication
4.3
Prove that dim
(
V
)
≤
1 if and only if the only subspaces of a vector space
V
are
{
0
}
and
V
itself.
Solution.
Assume that dim
(
V
)
=
1 (the result is obvious if dim
(
V
)
=
0).
Let
e
be a basis of
V
, let
{
0
} =
S
⊆
V
, and let
s
=
0 lie in
S
. The list
e
,
s
must be linearly dependent, so that there are scalars
α, β
, not both 0, with
α
e
+
β
s
=
0. Now
α
=
0, lest
β
s
=
0 and
β
=
0 (for
s
=
0). Hence,
e
=
α
−
1
β
s
∈
S
, forcing
V
⊆
S
, and so
S
=
V
.
Conversely, suppose the only subspaces of
V
are
{
0
}
and
V
. Assume
V
= {
0
}
(otherwise dim
(
V
)
=
0 and we are done), and choose
v
∈
V
with
v
=
0. If
u
∈
V
, then
v
=
V
=
v,
u
(for neither subspace is
{
0
}
), so
that
v,
u
is a linearly dependent list. Therefore, dim
(
V
)
=
1.
4.4
Prove, in the presence of all the other axioms in the de
fi
nition of vector
space, that the commutative law for vector addition is redundant; that is, if
V
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 KeithCornell
 Linear Algebra, Vector Space, linearly independent list, Matn

Click to edit the document details