Before discussing the implications of the axioms, we give some examples. We shall refer to these examples over and over again, and we shall use the notation established here throughout the rest of our work.
Example 1. Let \(\mathbb{C}^{1}\) ( \(=\mathbb{C}\) ) be the set of all complex numbers; if we interpret \(x+y\) and \(\alpha x\) as ordinary complex numerical addition and multiplication, \(\mathbb{C}^{1}\) becomes a complex vector space.
Example 2. Let \(\mathcal{P}\) be the set of all polynomials, with complex coefficients, in a variable \(t\) . To make \(\mathcal{P}\) into a complex vector space, we interpret vector addition and scalar multiplication as the ordinary addition of two polynomials and the multiplication of a polynomial by a complex number; the origin in \(\mathcal{P}\) is the polynomial identically zero.
Example (1) is too simple and example (2) is too complicated to be typical of the main contents of this book. We give now another example of complex vector spaces which (as we shall see later) is general enough for all our purposes.
Example 3. Let \(\mathbb{C}^{n}\) , \(n=1,2, \ldots\) , be the set of all \(n\) -tuples of complex numbers. If \(x=(\xi_{1}, \ldots, \xi_{n})\) and \(y=(\eta_{1}, \ldots, \eta_{n})\) are elements of \(\mathbb{C}^{n}\) , we write, by definition, \begin{align} x+y & =(\xi_{1}+\eta_{1}, \ldots, \xi_{n}+\eta_{n}) \\ \alpha x & =(\alpha \xi_{1}, \ldots, \alpha \xi_{n}) \\ 0 & =(0, \ldots, 0) \\ -x & =(-\xi_{1}, \ldots,-\xi_{n}). \end{align} It is easy to verify that all parts of our axioms (A) , (B) , and (C) , Section: Vector spaces , are satisfied, so that \(\mathbb{C}^{n}\) is a complex vector space; it will be called \(\mathbf{n}\) -dimensional complex coordinate space .
Example 4. For each positive integer \(n\) , let \(\mathcal{P}_{n}\) be the set of all polynomials (with complex coefficients, as in example (2)) of degree \(\leq n-1\) , together with the polynomial identically zero. (In the usual discussion of degree, the degree of this polynomial is not defined, so that we cannot say that it has degree \(\leq n-1\) .) With the same interpretation of the linear operations (addition and scalar multiplication) as in (2), \(\mathcal{P}_{n}\) is a complex vector space.
Example 5. A close relative of \(\mathbb{C}^{n}\) is the set \(\mathbb{R}^{n}\) of all \(n\) -tuples of real numbers. With the same formal definitions of addition and scalar multiplication as for \(\mathbb{C}^{n}\) , except that now we consider only real scalars \(\alpha\) , the space \(\mathbb{R}^{n}\) is a real vector space; it will be called \(\mathbf{n}\) -dimensional real coordinate space .
Example 6. All the preceding examples can be generalized. Thus, for instance, an obvious generalization of (1) can be described by saying that every field may be regarded as a vector space over itself. A common generalization of (3) and (5) starts with an arbitrary field \(\mathbb{F}\) and forms the set \(\mathbb{F}^{n}\) of \(n\) -tuples of elements of \(\mathbb{F}\) ; the formal definitions of the linear operations are the same as for the case \(\mathbb{F}=\mathbb{C}\) .
Example 7. A field, by definition, has at least two elements; a vector space, however, may have only one. Since every vector space contains an origin, there is essentially (i.e., except for notation) only one vector space having only one vector. This most trivial vector space will be denoted by \(\mathcal{O}\) .
Example 8. If, in the set \(\mathbb{R}\) of all real numbers, addition is defined as usual and multiplication of a real number by a rational number is defined as usual, then \(\mathbb{R}\) becomes a rational vector space.
Example 9. If, in the set \(\mathbb{C}\) of all complex numbers, addition is defined as usual and multiplication of a complex number by a real number is defined as usual, then \(\mathbb{C}\) becomes a real vector space. (Compare this example with (1); they are quite different.)