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204 ALGEBRA

3 . Relations between solutions of the nonhomogeneous system (5.5.1.1) and solutions of
◦
the corresponding homogeneous system (5.5.1.3).
1. The sum of any solution of the nonhomogeneous system (5.5.1.1) and any solution of
the corresponding homogeneous system (5.5.1.3) is a solution of system (5.5.1.1).
2. The difference of any two solutions of the nonhomogeneous system (5.5.1.1) is a solution
of the homogeneous system (5.5.1.3).
3. The sum of a particular solution X 0 of the nonhomogeneous system (5.5.1.1) and the
general solution (5.5.2.5) of the corresponding homogeneous system (5.5.1.3) yields the
general solution X of the nonhomogeneous system (5.5.1.1).

5.6. Linear Operators

5.6.1. Notion of a Linear Operator. Its Properties

5.6.1-1. Definition of a linear operator.
An operator A acting from a linear space V of dimension n to a linear space W of dimension
m is a mapping A : V→ W that establishes correspondence between each element x of the
space V and some element y of the space W. This fact is denoted by y = Ax or y = A(x).
An operator A : V→ W is said to be linear if for any elements x 1 and x 2 of the space
V and any scalar λ, the following relations hold:

A(x 1 + x 2 )= Ax 1 + Ax 2 (additivity of the operator),
A(λx)= λAx (homogeneity of the operator).

A linear operator A : V→ W is said to be bounded if it has a finite norm,which is
defined as follows:
Ax
A =sup =sup Ax ≥ 0.
x V x x =1
x ≠0
Remark. If A is a linear operator from a Hilbert space V into itself, then
|(x, Ay)|
Ax
A =sup =sup Ax =sup = sup |(x, Ay)|.
x V x x =1 x,y≠0 x y x = y =1
x ≠0
THEOREM. Any linear operator in a finite-dimensional normed space is bounded.
The set of all linear operators A : V→ W is denoted by L(V, W).
A linear operator O in L(V, W) is called the zero operator if it maps any element x of
V to the zero element of the space W: Ox = 0.
A linear operator A in L(V, V) is also called a linear transformation of the space V.
A linear operator I in L(V, V) is called the identity operator if it maps each element x
of V into itself: Ix = x.

5.6.1-2. Basic operations with linear operators.

The sum of two linear operators A and B in L(V, W) is a linear operator denoted by A + B
and defined by
(A + B)x = Ax + Bx for any x V.
The product of a scalar λ and a linear operator A in L(V, W) is a linear operator denoted
by λA and defined by
(λA)x = λAx for any x V.