2
Linear independence/Bases and Dimension
Vectors
±v
1
,±v
2
, . . . ,±v
k
in a
vector space
V
are
linearly dependent
if there are scalars
c
1
, c
2
, . . . , c
k
so
±
0 =
c
1
±v
1
+
c
2
±v
2
+
···
+
c
k
±v
k
,
with at least one
c
j
±
= 0
.
Such a vector
equation is said to be a
relation
of linear dependence among the
±v
1
,±v
2
, . . . ,±v
k
.
A collection of vectors for which the
only solution of the vector equation is
the
trivial solution,
c
1
= 0
, . . . , c
k
= 0
is said to be
linearly independent.
Examples: (1)
±
i,
±
j
in
R
2
; and
(2)
{
±
i,
±
j,
±
k
}
in
R
3
; are linearly independent.
Veriﬁcation:
c
1
±
i
+
c
2
±
j
+
c
3
±
k
=
c
1
(1
,
0
,
0) +
c
2
(0
,
1
,
0) +
c
3
(0
,
0
,
1) = (
c
1
, c
2
, c
3
)
= (0
,
0
,
0) =
±
0
,
only when
c
1
= 0
, c
2
= 0
, c
3
= 0
.