1
For a finite subset of a vector space, the definitions of L.D. and L.I.
are the same as those in
R
n
.
Example:
S
= {
x
2
−
3
x
+ 2, 3
x
2
−
5
x
, 2
x
−
3} of
P
2
is L.D., since
Example:
of
R
2
×
2
is L.I., since
implies that
a
=
b
=
c
= 0.
Example:
S
= {
e
t
,
e
2
t
,
e
3
t
} is a L.I. subset of
F
(
R
), since if
h
(
t
) =
ae
t
+
be
2
t
+
ce
3
t
= 0
∀
t
, then
h
(0) =
a
+
b
+
c
=0,
h
′
(0) =
a
+ 2
b
+ 3
c
= 0, and
h
′′
(0) =
a
+ 4
b
+ 9
c
= 0 together
imply
a
=
b
=
c
= 0.
Example: The infinite subset {1,
x
,
x
2
,
"
,
x
n
,
"
} of
P
is L.I., since
given any nonempty index set
I
,
Σ
i
∈
I
c
i
x
i
= 0 implies
c
i
= 0
for all
i
∈
I
.
Thus its every nonempty finite subset is L.I.
.
Example: The infinite subset {1+
x
, 1
−
x
, 1+
x
2
, 1
−
x
2
,
"
,1+
x
n
, 1
−
x
n
,
"
}
of
P
is L.D., since it contains L.D. finite subsets like
{1+
x
, 1
−
x
, 1+
x
2
, 1
−
x
2
}, in which
1(1+
x
) + 1(1
−
x
) + (
−
1)(1+
x
2
) + (
−
1)( 1
−
x
2
)=
0
.
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Proof
c
1
T
(
v
1
) +
c
2
T
(
v
2
) +
"
+
c
k
T
(
v
k
) =
0
⇒
T
(
c
1
v
1
+
c
2
v
2
+
"
+
c
k
v
k
) =
0
⇒
c
1
v
1
+
c
2
v
2
+
"
+
c
k
v
k
=
0
because
T
is onetoone
⇒
c
1
=
c
2
=
"
=
c
k
= 0.
8
Definition.
A subset
S
of a vector space
V
is a
basis of
V
if
S
is a L.I.
set and a generating set of
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 Spring '09
 Fong
 Linear Algebra, Vector Space, Axiom of choice, ΦB

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