1.5.
1.5. Linear Dependence and Linear Independence
1
.
(a) If
S
is a linearly dependent set, then each vector in
S
is a linear combina-
tion of other vector in
S
.
Ans
: F
(Example)
V
=
R
3
,
S
=
{
e
1
= (1
,
0
,
0)
,e
2
= (0
,
1
,
0)
,e
3
= (0
,
0
,
1)
,e
4
= (1
,
1
,
0)
}
{
e
1
,e
2
,e
3
}
: linearly independent
{
e
1
,e
2
,e
4
}
: linearly dependent
(b) Any set containing the zero vector is linearly dependent.
Ans
: T (
∵
)
∀
a
∈
F,
0 =
a
·
0
, a
6
= 0
(c) The empty set is linearly dependent.
Ans
: F (
∵
) linearly dependent set must be non-empty.
(d) Subsets of linearly dependent sets are linearly dependent.
Ans
: F (
∵
) theorem 1.6
(e) Subsets of linearly independent sets are linearly independent.
Ans
: T (
∵
) the corollarly from theorem 1.6
(f) If
a
1
x
1
+
a
2
x
2
+
···
+
a
n
x
n
= 0 and
x
1
,x
2
,
···
,x
n
are linearly independent,
then all the scalars
a
i
are zero.
Ans
: T (
∵
) from the deﬁnition.
2
.
(a), (d), (e), (g), (h), (j) : linearly independent
(b), (c), (f), (i) : linearly dependent
24
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