Cryptography Reference
In-Depth Information
If
is not linearly independent, then it is called
linearly dependent
. A linearly
independent subset of a vector space that spans
V
is called a
basis
for
V
. The
number of elements in a basis is called the
dimension
of
V
.A
hyperplane
H
is
an
(
n
S
−
1)
-dimensional subspace of an
n
-dimensional vector space
V
.
Example A.11
For a given prime
p
,
m,n
∈
N
, the finite field
F
p
n
is an
n
-
F
p
m
with
p
mn
elements.
dimensional vector space over
A.6 Basic Matrix Theory
n
matrix (read “
m
by
n
matrix”) is a rectangular
array of entries with
m
rows and
n
columns. For simplicity, we will assume that
the entries come from a field
F
.If
A
is such a matrix, and
a
i,j
denotes the entry
in the
i
th row and
j
th column, then
If
m,n
∈
N
, then an
m
×
a
1
,
1
a
1
,
2
···
a
1
,n
a
2
,
1
a
2
,
2
···
a
2
,n
A
=(
a
i,j
)=
.
.
.
.
a
m,
1
a
m,
2
···
a
m,n
n
matrices
A
=(
a
i,j
), and
B
=(
b
i,j
) are equal if and only if
a
i,j
=
b
i,j
for all
i
and
j
. The matrix (
a
j,i
) is called the
transpose
of
A
, denoted
by
Two
m
×
A
t
=(
a
j,i
)
.
Addition of two
m
×
n
matrices
A
and
B
is done in the natural way.
A
+
B
=(
a
i,j
)+(
b
i,j
)=(
a
i,j
+
b
i,j
)
,
F
, then
rA
=
r
(
a
i,j
)=(
ra
i,j
), called
scalar multiplication
.
Matrix products are defined by the following.
If
A
=(
a
i,j
)isan
m
∈
and if
r
×
n
matrix and
B
=(
b
j,k
)isan
n
×
r
matrix, then the
product
of
A
and
B
is defined as the
m
×
r
matrix:
AB
=(
a
i,j
)(
b
j,k
)=(
c
i,k
)
,
where
n
c
i,k
=
a
i,
b
,k
.
=1
Multiplication, if defined, is associative, and distributive over addition. If
m
=
n
, then
1
F
0
···
0
0
F
···
0
I
n
=
.
.
.
.
00
···
1
F
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