Examples

Example 1. Let ( ) be the set of all complex numbers; if we interpret and as ordinary complex numerical addition and multiplication, becomes a complex vector space.

Example 2. Let be the set of all polynomials, with complex coefficients, in a variable . To make 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 is the polynomial identically zero.

Example 3. Let , , be the set of all -tuples of complex numbers. If and are elements of , we write, by definition, It is easy to verify that all parts of our axioms (A) , (B) , and (C) , Section: Vector spaces , are satisfied, so that is a complex vector space; it will be called -dimensional complex coordinate space .

Example 4. For each positive integer , let be the set of all polynomials (with complex coefficients, as in example (2)) of degree , 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 .) With the same interpretation of the linear operations (addition and scalar multiplication) as in (2), is a complex vector space.

Example 5. A close relative of is the set of all -tuples of real numbers. With the same formal definitions of addition and scalar multiplication as for , except that now we consider only real scalars , the space is a real vector space; it will be called -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 and forms the set of -tuples of elements of ; the formal definitions of the linear operations are the same as for the case .

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 .

Example 8. If, in the set of all real numbers, addition is defined as usual and multiplication of a real number by a rational number is defined as usual, then becomes a rational vector space.

Example 9. If, in the set of all complex numbers, addition is defined as usual and multiplication of a complex number by a real number is defined as usual, then becomes a real vector space. (Compare this example with (1); they are quite different.)