We come now to the objects that really make vector spaces interesting.
Definition 1. A linear transformation (or operator ) \(A\) on a vector space \(v\) is a correspondence that assigns to every vector \(x\) in \(\mathcal{V}\) a vector \(A x\) in \(\mathcal{V}\) , in such a way that \[A(\alpha x+\beta y)=\alpha A x+\beta A y\] identically in the vectors \(x\) and \(y\) and the scalars \(\alpha\) and \(\beta\) .
We make again the remark that we made in connection with the definition of linear functionals, namely, that for a linear transformation \(A\) , as we defined it, \(A 0=0\) . 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 \(0\) and \(1\) respectively, are defined (for all \(x\) ) by \(0 x=0\) and \(1 x=x\) .
Example 2. Let \(x_{0}\) be any fixed vector in \(\mathcal{V}\) , and let \(y_{0}\) be any linear functional on \(\mathcal{V}\) ; write \(A x=y_{0}(x) \cdot x_{0}\) . More generally: let \(\{x_{1}, \ldots, x_{n}\}\) be an arbitrary finite set of vectors in \(v\) and let \(\{y_{1}, \ldots, y_{n}\}\) be a corresponding set of linear functionals on \(\mathcal{V}\) ; write \(A x=y_{1}(x) x_{1}+\cdots+y_{n}(x) x_{n}\) . It is not difficult to prove that if, in particular, \(\mathcal{V}\) is \(n\) -dimensional, and the vectors \(x_{1}, \ldots, x_{n}\) form a basis for \(\mathcal{V}\) , then every linear transformation \(A\) has the form just described.
Example 3. Let \(\pi\) be a permutation of the integers \(\{1, \ldots, n\}\) ; if \(x=(\xi_{1}, \ldots, \xi_{n})\) is a vector in \(\mathbb{C}^{n}\) , write \(A x=(\xi_{\pi(1)}, \ldots, \xi_{\pi(n)})\) . Similarly, let \(\pi\) be a polynomial with complex coefficients; if \(x\) is a vector (polynomial) in \(\mathcal{P}\) , write \(A x=y\) for the polynomial defined by \(y(t)=x(\pi(t))\) .
Example 4. For any \(x\) in \(\mathcal{P}_n, x(t)=\sum_{j=0}^{n-1} \xi_{j} t^{j}\) , write \((D x)(t)=\sum_{j=0}^{n-1} j \xi_{j} t^{j-1}\) . (We use the letter \(D\) here as a reminder that \(D x\) is the derivative of the polynomial \(x\) . We remark that we might have defined \(D\) on \(\mathcal{P}\) as well as on \(\mathcal{P}_n\) ; 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 \(x\) in \(\mathcal{P}\) , \(x(t)=\sum_{j=0}^{n-1} \xi_{j} t^{j}\) , write \((Sx)(t)=\sum_{j=0}^{n-1} \frac{\xi_{j}}{j+1} t^{j+1}\) . (Once more we are disguising by algebraic notation a well-known analytic concept. Just as in (4) \((D x)(t)\) stood for \(\frac{d x}{d t}\) , so here \((S x)(t)\) is the same as \(\displaystyle \int_{0}^{t} x(s) \,d s.\) )
Example 6. Let \(m\) be a polynomial with complex coefficients in a variable \(t\) . (We may, although it is not particularly profitable to do so, consider \(m\) as an element of \(\mathcal{P}\) .) For every \(x\) in \(\mathcal{P}\) , we write \(M x\) for the polynomial defined by \((M x)(t)=m(t) x(t)\) . For later purposes we introduce a special symbol; in case \(m(t)=t\) , we shall write \(T\) for the transformation \(M\) , so that \((T x)(t)=t x(t)\) .