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5.3. LINEAR SPACES 187

The Cayley–Hamilton theorem can also be used to find the powers and the inverse of a
k
matrix A (since if f A (A)= 0,then A f A (A)= 0 for any positive integer k).
Example 8. For the matrix in Examples 4–7, one has
3
2
f A(A)=–A – 6A – 11A – 6I = 0.
Hence we obtain 3 2
A =–6A – 11A – 6I.
By multiplying this expression by A, we obtain
4 3 2
A =–6A – 11A – 6A.
Now we use the representation of the cube of A via lower powers of A and eventually arrive at the formula
4 2
A = 25A + 60A + 36I.
For the inverse matrix, by analogy with the preceding, we obtain
–1 –1 3 2 2 –1
A f A(A)= A (–A – 6A – 11A – 6I)=–A – 6A – 11I – 6A = 0.
The definitive result is 1
–1
2
A =– (A + 6A + 11I).
6
In some cases, an analytic function of a matrix A can be computed by a formula in the
following theorem.
SYLVESTER’S THEOREM. If all eigenvalues of a matrix A are distinct, then
n
(A – λ i I)
f(A)= f(λ k )Z k , Z k = i≠k ,
k=1 i≠k (λ k – λ i )
and, moreover, Z k = Z k m (m = 1, 2, 3, ...).

5.3. Linear Spaces
5.3.1. Concept of a Linear Space. Its Basis and Dimension

5.3.1-1. Definition of a linear space.
A linear space or a vector space over a field of scalars (usually, the field of real numbers
or the field of complex numbers) is a set V of elements x, y, z, ... (also called vectors)of
any nature for which the following conditions hold:
I. There is a rule that establishes correspondence between any pair of elements x, y V
and a third element z V, called the sum of the elements x, y and denoted by z = x + y.
II. There is a rule that establishes correspondence between any pair x, λ,where x is an
element of V and λ is a scalar, and an element u V, called the product of a scalar λ
and a vector x and denoted by u = λx.
III. The following eight axioms are assumed for the above two operations:
1. Commutativity of the sum: x + y = y + x.
2. Associativity of the sum: (x + y)+ z = x +(y + z).
3. There is a zero element 0 such that x + 0 = x for any x.
4. For any element x there is an opposite element x such that x + x = 0.

5. A special role of the unit scalar 1: 1 ⋅ x = x for any element x.
6. Associativity of the multiplication by scalars: λ(μx)=(λμ)x.
7. Distributivity with respect to the addition of scalars: (λ + μ)x = λx + μx.
8. Distributivity with respect to a sum of vectors: λ(x + y)= λx + λy.
This is the definition of an abstract linear space. We obtain a specific linear space if
the nature of the elements and the operations of addition and multiplication by scalars are
concretized.