4 1. Vecto r Spaces Hint: Yo u may use the following fact: if p is a prime, and both n and m are not divisible by p, then nm is not divisible by p. Sho w that this implies that if n ^ 0 in Zp, then {nm : m G Zp} covers all of Zp. 1.2 Vecto r space s 1.2.1 DEFINITION . A vector space Y over a field ¥ is an abelian group (y,+), fo r whic h a binary product, (a,v ) ^ av , of F x f int o Y i s defined, satisfyin g the following axioms for all a, b G F and «,vGf : VS1. l v = v, VS2. (ofe) v = a(fev) , VS 3. ( a + fo)v = a v + bv an d a( v + w) = av + au. Other familiar properties may be derived from these. For example, for every v E f,O v = ( 1 - l) v = v - v = 0 . The elements of Y ar e usually referred to as vectors', the elements of the underlying field as scalars. Observe that in the equality (ab)v = a(bv) th e multiplication (ab) is within F the others are products of a vector by a scalar. In (a+b) v = av + bv, the addition on the left is the addition in F, while that on the right is the addition in Y. Most o f th e notion s an d result s w e discus s ar e vali d fo r vecto r spaces over arbitrary fields. When the underlying field does not need to be specified, we denote it by the generic F. Results that apply to vector spaces over specific fields, or to vector spaces over fields satisfying som e additional conditions, will be stated explicitly in terms of the appropriate field or the additional conditions. If the underlying field is R or C, then the vector space is called a real vector space or a complex vector space, respectively. Vector spaces may also have additional structures: geometric, such as inner-product, whic h we study in Chapter 6 o r algebraic, suc h as multiplication, as discussed in A.5.6. EXAMPLES : The following sets are all vector spaces over the indicated fields.

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