306
Appendix A
In the text, we will have occasion to refer to such items as vector spaces, so we remind the reader of the definition. The reader is referred to pages 77–81, where we discussed the axioms for algebraic objects such as groups, rings, and fields. In particular, for the sake of completeness, note that any set satisfying all of the axioms of Theorem 2.1 on page 77, except (g), is called a division ring. ✦ Vector Spaces A vector space consists of an additive abelian group V and a field F together with an operation called scalar multiplication of each element of V by each element of F on the left, such that for each r, s ∈ F and each α, β ∈ V the following conditions are satisfied: A.1. rα ∈ V . A.2. r(sα) = (rs)α. A.3. (r + s)α = (rα) + (sα). A.4. r(α + β) = (rα) + (rβ). A.5. 1F α = α. The set of elements of V are called vectors and the elements of F are called scalars. The generally accepted abuse of language is to say that V is a vector space over F . If V1 is a subset of a vector space V that is a vector space in its own right, then V1 is called a subspace of V . Example A.9 For a given prime p, m, n ∈ N, the finite field Fpn is an ndimensional vector space over Fpm with pmn elements.
Definition A.22 Bases, Dependence, and Finite Generation If S is a subset of a vector space V , then the intersection of all subspaces of V containing S is called the subspace generated by S, or spanned by S. If there is a finite set S, and S generates V , then V is said to be finitely generated. If S = ∅, then S generates the zero vector space. If S = {m}, a singleton set, then the subspace generated by S is said to be the cyclic subspace generated by m. A subset S of a vector space V is said to be linearly independent provided that for distinct s1 , s2 , . . . , sn ∈ S, and rj ∈ V for j = 1, 2, . . . , n, n &
rj sj = 0 implies that rj = 0 for j = 1, 2, . . . , n.
j=1
If S is not linearly independent, then it is called linearly dependent. A linearly independent subset of a vector space that spans V is called a basis for V .