Linear transformations

We come now to the objects that really make vector spaces interesting.

Definition 1. A linear transformation (or operator ) on a vector space is a correspondence that assigns to every vector in a vector in , in such a way that identically in the vectors and and the scalars and .

We make again the remark that we made in connection with the definition of linear functionals, namely, that for a linear transformation , as we defined it, . For this reason such transformations are sometimes called homogeneous linear transformations.

Before discussing any properties of linear transformations we give several examples. We shall not bother to prove that the transformations we mention are indeed linear; in all cases the verification of the equation that defines linearity is a simple exercise.

Example 1. Two special transformations of considerable importance for the study that follows, and for which we shall consistently reserve the symbols and respectively, are defined (for all ) by and .

Example 2. Let be any fixed vector in , and let be any linear functional on ; write . More generally: let be an arbitrary finite set of vectors in and let be a corresponding set of linear functionals on ; write . It is not difficult to prove that if, in particular, is -dimensional, and the vectors form a basis for , then every linear transformation has the form just described.

Example 3. Let be a permutation of the integers ; if is a vector in , write . Similarly, let be a polynomial with complex coefficients; if is a vector (polynomial) in , write for the polynomial defined by .

Example 4. For any in , write . (We use the letter here as a reminder that is the derivative of the polynomial . We remark that we might have defined on as well as on ; we shall make use of this fact later. Observe that for polynomials the definition of differentiation can be given purely algebraically, and does not need the usual theory of limiting processes.)

Example 5. For every in , , write . (Once more we are disguising by algebraic notation a well-known analytic concept. Just as in (4) stood for , so here is the same as )

Example 6. Let be a polynomial with complex coefficients in a variable . (We may, although it is not particularly profitable to do so, consider as an element of .) For every in , we write for the polynomial defined by . For later purposes we introduce a special symbol; in case , we shall write for the transformation , so that .

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