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5.6. LINEAR OPERATORS 205

The opposite operator for an operator A L(V, W) is an operator denoted by –A and
defined by
–A =(–1)A.
The product of two linear operators A and B in L(V, V) is a linear operator denoted by
AB and defined by
(AB)x = A(Bx) for any x V.
Properties of linear operators in L(V, V):
(AB)C = A(BC) (associativity of the product of three operators),
λ(AB)= (λA)B (associativity of multiplication of a scalar and two operators),
(A + B)C = AC + BC (distributivity with respect to the sum of operators),
where λ is a scalar; A, B,and C are linear operators in L(V, V).
Remark. Property 1 allows us to define the product A 1A 2 ... A k of finitely many operators in L(V, V)
and the kth power of an operator A,
k
A = AA ... A .

k times
The following relations hold:
pq
p
q
p q
A p+q = A A , (A ) = A . (5.6.1.1)
5.6.1-3. Inverse operators.
A linear operator B is called the inverse of an operator A in L(V, V)if AB = BA = I.The
–1
inverse operator is denoted by B = A . If the inverse operator exists, the operator A is said
to be invertible or nondegenerate.
k –1
–1 k
Remark. If A is an invertible operator, then A –k =(A ) =(A ) and relations (5.6.1.1) still hold.
A linear operator A from V to W is said to be injective if it maps any two different
elements of V into different elements of W, i.e., for x 1 ≠ x 2 ,we have Ax 1 ≠ Ax 2 .
If A is an injective linear operator from V to V, then each element y V is an image of
some element x V: y = Ax.
THEOREM. A linear operator A : V→ V is invertible if and only if it is injective.

5.6.1-4. Kernel, range, and rank of a linear operator.

The kernel of a linear operator A : V→ V is the set of all x in V such that Ax = 0.The
kernel of an operator A is denoted by ker A and is a linear subspace of V.
The range of a linear operator A : V→ V is the set of all y in V such that y = Ax.The
range of a linear operator A is denoted by im A and is a subspace of V.
Properties of the kernel, the range, and their dimensions:
1. For a linear operator A : V→ V in n-dimensional space V, the following relation holds:
dim (im A)+ dim (ker A)= n.

2. Let V 1 and V 2 be two subspaces of a linear space V and dim V 1 +dim V 2 =dim V.Then
there exists a linear operator A : V→ V such that V 1 =im A and V 2 =ker A.
A subspace V 1 of the space V is called an invariant subspace of a linear operator
A : V→ V if for any x in V 1 , the element Ax also belongs to V 1 . A linear operator A : V→ V
is said to be reducible if V can be represented as a direct sum V = V 1 ⊕ ··· ⊕ V N of two or
more invariant subspaces V 1 , ... , V N of the operator A,where N is a natural number.
Example 1. ker A and im A are invariant subspaces of any linear operator A : V→ V.