Many of the properties of
are shared by
For instance, the scalar multiplicative
identity is the scalar 1 and the additive identity in
is
The
standard
basis
for
is simply
which is the standard basis for
Because this basis contains
n
vectors, it follows that the
dimension of
is
n.
Other bases exist; in fact, any linearly independent set of
n
vectors in
can be used, as demonstrated in Example 2.
EXAMPLE 2
Verifying a Basis
Show that
is a basis for
Solution
Because
has a dimension of 3, the set
will be a basis if it is linearly inde-
pendent. To check for linear independence, set a linear combination of the vectors in
S
equal
to
0
as follows.
This implies that
So,
and you can conclude that
is linearly independent.
EXAMPLE 3
Representing a Vector in C
n
by a Basis
Use the basis
S
in Example 2 to represent the vector
v
5
s
2,
i
, 2
2
i
d
.
H
v
1
,
v
2
,
v
3
J
c
1
5
c
2
5
c
3
5
0,
c
3
i
5
0.
c
2
i
5
0
s
c
1
1
c
2
d
i
5
0
ss
c
1
1
c
2
d
i
,
c
2
i
,
c
3
i
d
5
s
0, 0, 0
d
s
c
1
i
,
0, 0
d
1
s
c
2
i
,
c
2
i
, 0
d
1
s
0, 0,
c
3
i
d
5
s
0, 0, 0
d
c
1
v
1
1
c
2
v
2
1
c
3
v
3
5
s
0, 0, 0
d
H
v
1
,
v
2
,
v
3
J
C
3
C
3
.
S