j
=1:
p
, the product
C
=
A
ú
B
= [
c
ij
]
i
=1:
m
,
j
=1:
p
is the matrix of size [
m
,
p
] with components
c
ik
=
n
ÿ
j
=1
a
ij
b
jk
,
for all
i
,
k
We typically omit the
ú
, and write
C
=
AB
for the matrix product.
3. Given
A
= [
a
ij
]
i
=1:
m
,
j
=1:
n
and a complex number
–
, the scalar multiple
C
=
–
A
is
the matrix of the same size with components
c
ij
=
–
a
ij
,
for all
i
,
j
Definition 1.
1. We have
zero matrices
O
m
,
n
of all sizes.
2. The
identity matrix
I
n
of size [
n
,
n
]
(a “square” matrix) has components
1
on the
diagonal, and
0
o
ff
the diagonal.
3. Some, but not all, square matrices
A
have an
inverse
A
≠
1
such that
A
≠
1
A
=
AA
≠
1
=
I
n
: when the inverse exists,
A
is called an
invertible matrix
.

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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
ú
= [
a
ú
ij
]
i
=1:
m
,
j
=1:
n
with
a
ú
ij
= (
a
ij
)
ú
for all
i
,
j
.
6. The
Hermitian conjugate
of
A
is the matrix
A
†
= (
A
T
)
ú
= (
A
ú
)
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
–
,
—
,
“
,
. . .
:
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
–
,
—
,
· · ·
œ
C
, which is associative
–
(
—˛
a
) = (
–—
)
˛
a
and distributive
–
(
˛
a
+
˛
b
) =
–˛
a
+
–
˛
b
,
(
–
+
—
)
˛
a
=
–˛
a
+
—˛
a
.
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
=
Œ
,
W
N
, the set of complex trigonometric polynomials
of the form
f
(
x
) =
q
N
n
=
≠
N
a
n
e
inx
where
a
n
œ
C
for each
n
.
5. “Standard”
R
n
, the set of real
[
n
, 1]
matrices
X
(i.e. column
n
-vectors), is likewise
the first example of a
real vector space
, i.e a vector space taken with real scalars.

8
CHAPTER 1.
LINEAR ALGEBRA (APPROX. 9 LECTURES)
The central theme here is that many important examples exist which are not naturally
a
C
n
or
R
n
.

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