If a square matrix A is invertible, its rows (or columns) are linearly independent.
In this case, the linear equation system A.x = 0 with x =(x
1
,…,x
m
)
T
has only the tri-
vial solution x = 0.IfA is singular, i.e., rows (or columns) are linearly dependent,
then the linear equation system A.x = 0 has a nontrivial solution.
The determinant of A (DetA) is a real or complex number that can be assigned to
every square matrix. For the 161 matrix (a
11
), it holds that DetA = a
11
.Fora262
matrix, it is calculated as
Det
a
11
a
12
a
21
a
22
a
11
a
12
a
21
a
22
a
11
a
22
a
12
a
21
: (3-25)
The value of a determinant of higher order can be obtained by an iterative proce-
dure, i.e., by expanding the determinant with respect to one row or column: sum up
every element of this row (or column) multiplied by the value of its adjoint. The ad-
joint A
ik
of element a
ik
is obtained by deleting the i-th row and the k-th column of
the determinant (forming the (i,k) minor of A), calculating the value of the (i,k)
minor and multiplying by (–1)
i+k
. For example, a determinant of third order is
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
a
11
A
11
a
12
A
12
a
13
A
13
a
11
1
2
a
22
a
23
a
32
a
33
a
12
1
3
a
21
a
23
a
31
a
33
a
13
1
4
a
21
a
22
a
31
a
32
(3-26)
a
11
a
22
a
33
a
23
a
32
a
12
a
21
a
33
a
23
a
31
a
13
a
21
a
32
a
22
a
31
:
The value of a determinant is zero (1) if it contains a zero row or a zero column or
(2) if one row (or column) is a linear combination of the other rows (or columns). In
this case, the respective matrix is singular.
3.1.2.4 Dimension and Rank
Subspace of a vector space: Let us further consider the vector space V
n
of all n-di-
mensional column vectors (n 61). A subset S of V
n
is called a subspace of V
n
(1) if
the zero vector belongs to S, (2) if with two vectors belonging to S their sum also be-
longs to S, and (3) if with one vector belonging to S its scalar multiples also belong
to S. Vectors x
1
,…,x
m
belonging to a subspace S form a basis of a vector subspace S
if they are linearly independent and if every vector in S is a linear combination of x
1
,
…, x
m
. A subspace where at least one vector is nonzero has a basis. In general, a sub-
space will have more than one basis. Every linear combination of basis vectors is it-
self a basis. The number of vectors making up a basis is called the dimension of S
(dimS). For the n-dimensional vector space, it holds that dimS ^ n.
The rank of a matrix is an important integer associated with a matrix A of size
m6n. RankA is equal to the number of linearly independent columns or rows in A
and equal to the number of nonzero rows in the reduced row-echelon form of the
matrix A. It holds that Rank A ^ m,n.
64
3 Mathematics in a Nutshell