1.2.
VECTOR SPACES
7
4. The
transpose
of
A
= [
a
ij
]
i
=1:
m
,
j
=1:
n
is the matrix
A
T
= [
a
T
ij
]
i
=1:
n
,
j
=1:
m
with
a
T
ij
=
(
a
ji
)
for all
i
,
j
.
5. The
complex conjugate
of
A
= [
a
ij
]
i
=1:
m
,
j
=1:
n
is the matrix
A
*
= [
a
*
ij
]
i
=1:
m
,
j
=1:
n
with
a
*
ij
= (
a
ij
)
*
for all
i
,
j
.
6. The
Hermitian conjugate
of
A
is the matrix
A
†
= (
A
T
)
*
= (
A
*
)
T
.
Another very important property of matrix algebra, often overlooked in elementary
treatments, is that it respects
block decompositions
.
For example, one can think of a
[4, 6] matrix as a [2, 3] matrix whose components are [2, 2] matrices.
Moreover, matrix
multiplication can be performed in any way that respects such a block decomposition.
We will see examples of this in the following discussion.
1.2
Vector Spaces
We zoom straight to complex vector spaces.
Definition 2.
A set of objects
a
,
b
,
c
,
. . .
called
vectors
is called a
complex vector space
V
if for all vectors
a
,
b
,
c
,
. . .
and
scalars
α
,
β
,
γ
,
. . .
:
1. the set is closed under a vector addition with associativity
a
+ (
b
+
c
) = (
a
+
b
) +
c
and commutativity
a
+
b
=
b
+
a
.
2. the set is closed under multiplication by scalars
α
,
β
,
· · · ∈
C
, which is associative
α
(
βa
) = (
αβ
)
a
and distributive
α
(
a
+
b
) =
αa
+
αb
,
(
α
+
β
)
a
=
αa
+
βa
.
3. there is a zero vector
0
such that
0 +
a
=
a
.
4. the scalar
1
is such that
1
a
=
a
.
5. all vectors
a
have a negative

a
such that
a
+ (

a
) = 0
.
Here are some important examples of complex vector spaces, and one example of a
real vector space:
Examples 1.
1. “Standard”
C
n
, which we identify as the set of complex
[
n
, 1]
matrices
X
(i.e. column
n
vectors).
2. For each pair
m
,
n
of positive integers, the set of complex
[
m
,
n
]
matrices.
3.
C
[0, 1]
, the set of continuous functions from the interval
[0, 1]
to
C
.
4. For each
N
, including
N
=
∞
,
W
N
, the set of complex trigonometric polynomials
of the form
f
(
x
) =
∑
N
n
=

N
a
n
e
inx
where
a
n
∈
C
for each
n
.