Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 75

Definition 4.7 Let . The rank of T is the dimension of the range TV of T. It is denoted by r(T).

Definition 4.8 Let . The null space (or kernel) of T is the set of vectors x such that . It is denoted by N(T).

Definition 4.9 The dimension of the subspace N(T) is called the nullity and is denoted by n(T).

For example: the zero operator ω has and the identity operator i has .

Theorem 4.3 Let . Then .

We are now interested in determining in how far two vector spaces are ‘similar’ in some sense.

Theorem 4.4 Let . Then

 T is onto W (surjective) if and only if .

 T is one-to-one (injective) if and only if .

Definition 4.10 A linear transformation is called an isomorphism if it is both injective and surjective.

It is possible to form the sum of linear transformations and to compose linear transformations, and we discuss this topic in Chapter 5.