1
Lecture 12
12.1 Vector spaces with complex scalars
A complex vector space
is a set V of vectors together with vector addition
w
v
in V and
scalar multiplication
v
r
in V by elements of C. Moreover, for all the vectors in V, and scalars in
C, the following properties are satisfied:
Properties of vector addition:
A1: (
u
+
v
) +
w
=
u
+ (
v
+
w
)
an associatelaw
A2:
u
+
w
=
w
+
u
a commutative law
A3:
0
+
v
=
v
0
as additive identity
A4:
v
+ (
v
) =
0

v
as additive inverse of
v
Properties of vector scalar multiplication:
S1: r (
v
+
w
) = (r
v
) + (r
w
)
a distributive law
S2: (r + s)
v
=( r
v
) + (s
v
)
a distributive law
S3: r (s
v
) = (rs)
v
an associative law
S4: 1
v
=
v
preservation of scale
An example of complex vector space is the Euclidean space C
n
consisting of all vectors
of the form
n
u
u
u
u
...,
,
,
2
1
, where
i
u
C.
Example: Consider the set V of all matrix of the form
a
a
0
0
where
a
is in C, with matrix addition and scalar multiplication. Determine
whether V forms a complex vector space.
Example: The set of m
n matrices with complex entries and the operations of matrix
addition and scalar multiplication is a complex vector space.
Example: If f
1
(x) and f
2
(x) are realvalued functions of real variable x, then the expression
f(x) = f
1
(x) + if
2
(x) is called a complexvalued function of the real variable x. The set
of all f(x) with usual operations is a complex vector space.
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 Fall '09
 YANG
 Linear Algebra, Algebra, Addition, Multiplication, Vectors, Scalar, Vector Space, Euclidean space, Hilbert space, complex vector space

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