In this chapter we consider linear mappings over vector spaces. We begin by stating the definition and a discussion of the structural properties of linear mappings. We then introduce the notion of adjoint mappings and illustrate some of their applications. We next focus on linear mappings from a vector space into itself and study a series of important concepts such as invariance and reducibility, eigenvalues and eigenvectors, projections, nilpotent mappings, and polynomials of linear mappings. Finally we discuss the use of norms of linear mappings and present a few analytic applications.

Linear mappings

A linear mapping may be regarded as the simplest correspondence between two vector spaces. In this section we start our study with the definition of linear mappings. We then discuss the matrix representation of a linear mapping, composition of linear mappings, and the rank and nullity of a linear mapping.

2.1.1 Definition, examples, and notion of associated matrices

Let U and V be two vector spaces over the same field F. A linear mapping or linear map or linear operator is a correspondence T from U into V, written as T : U → V, satisfying the following.

1. (Additivity) T(u1 + u2) = T(u1) + T(u2), u1, u2 ∈ U.

2. (Homogeneity) T(au) = aT(u), a ∈ F, u ∈ U.

A special implication of the homogeneity condition is that T(0) = 0. One may also say that a linear mapping ‘respects’ or preserves vector addition and scalar multiplication.

The set of all linear mappings from U into V will be denoted by L(U, V). For S, T ∈ L(U, V), we define S + T to be a mapping from U into V Satisfying

(S + T)(u) = S(u) + T(u), ∀u ∈ U.

For any a ∈ F and T ∈ L(U, V), we define aT to be the mapping from U into V satisfying

(aT)(u) = aT(u), ∀u ∈ U.