Linear Algebra and Geometry

CHAPTER II. LINEAR TRANSFORMATIONS

CHAPTER I1 LINEAR TRANSFORMATIONS At the beginning of the last chapter, we gave a brief description of abstract algebra as the mathematical theory of algebraic systems and, in particular, linear algebra as the mathematical theory of linear spaces. These descriptions are incomplete, for we naturally want to find relations among the algebraic systems in question. In other words, we also have to study mappings between algebraic systems compatible with the algebraic structure in question. A linear space consists, by definition, of two parts: namely a non-empty set and an algebraic structure on this set. It is therefore natural to compare one linear space with another by means of a mapping of the underlying sets on theone hand. On the other hand, such a mapping should also take into account the algebraic structures . In other words we shall study mappings between the underlying sets of linear spaces that preserve linearity. These mappings are called linear transformations or homomorphisms and they will be the chief concern of the present chapter. 55.General Properties of Linear Transformation A. Linear transformation and examples In assigning coordinates to vectors of an n-dimensional linear space X over A relative to a base (X,, .฀ ..฀ .฀,฀X,), we obtained in §2E a mapping @:฀X +฀ An such that, for every vector X of X, As a mapping of the set X into the set An,Qi,฀ is bijective. Relative to the algebraic structure of the linear space X and of the linear space An, has the following properties: (i) @(X+฀Y)=฀@(X)+฀@฀W,฀ (ii) @(?U)฀ =฀A@฀(X), forany two vectors x and y of X and any scalar h of A. Note that X฀+฀y 46 II LINEARTRANSFORMATIONS and Xx฀ are formed according to the addition and the scalar multiplication of the linear space X, while +(X) +฀@(y) and h+(x) according to those of the linear space An.Since the algebraicstructure of a linear space is defined by its addition and scalar multiplication, the relations (i) and (ii) express that the mapping @฀is compatible with the algebraic structure of linear space. Therefore Q, is an example of the type of mapping that we are looking for. DEFINITION 5.1. Let X and Y be linear spaces over the same A. A mapping Q, of the set X into the set Y is called a linear transformation of the linear space X into the linear space Y if and only if for any vectors X and y of X and any scalar h of A, the equations hoId. Note that the domain and the range of a linear transformation must be linear spaces over the same A. In other words, we do not consider as a linear transformation any mapping of a real linear space into a complex linear space even if (i) holds and (ii) also holds for all XER. Therefore, whenever we say: '@:฀ X +฀Y is a linear transforrnation ' we mean that X and Y are linear spaces over the same A and Q, is a linear transformation of the linear space X into the linear space Y. Since sums and scalar products are expressible as linear combinations , we can replace conditions (i) and (ii) by a single equivalent condition (iii) @(h +฀py) =฀hQ,(x)+฀pQ,(y)฀for any X,฀y EX and h, p฀EA. Property (iii) is called linearity of Q,. For linear spaces over A, linear transformation, linear mapping, A-homomorphism, and homomorphism are synonymous. EXAMPLE 5.2. Let X be a linear space over A and Y a subspace of X. Then the inclusion mapping 6:฀ Y -t฀X is clearly an injeclive linear transformation. In particular, the identity mapping ix: X -t฀X is a bijective linear transformation. EXAMPLE 5.3. Let X and Y be two linear spaces over the same A. Then the constant mapping 0:฀X +฀Y, such that O(x) =฀0฀for every XEX,is a linear transformation called the zero linear transformation. It is easily seen that the zero linear mapping is the only constant linear transformation of the linear space X into the linear space Y. 5 5 ฀ GENERAL PROPERTIES OF LINEAR TRANSFORMATION 47฀ EXAMPLE 5.4. Let Y be a subspace of a linear space X and X / Y the quotient space of X by Y . Then the natural surjection q: X +฀ X / Y defined by q(x) =฀[ X ] for every xcX is a surjective linear transformation . EXAMPLE 5.5. If @:฀ X...