Chapter 1 Vector Spaces 1.1 Group s and fields Vector space s ar e defined ove r fields, and the definitions o f both fields and vector spaces depend on the notion of a commutative group. Groups and fields are reviewed in more detail in the appendix but, for convenience, we include here their definitions alon g with some basic examples. 1.1.1 Groups . DEFINITION: A group is a pair (G , *), wher e G is a se t an d * is a binary operation (x,y) \— x*y, define d fo r al l pairs (x,y) G Gx G , taking values in G, and satisfying th e following conditions: G-l Th e operation is associative: For all x,y,z G G, (x * y) * z x * (j * z). G-2 Ther e exists a unique element e £ G, called the identity element or the unit of G, such that e*x = x*e = xfor all * G G. G-3 Fo r ever y * G G there exist s a unique elemen t x _1 , calle d the inverse of x, such that x~l *x = x*x~l = e. A grou p (G , *) i s abelian, o r commutative, i f **)/ = y *x fo r al l x and y. Th e group operation in a commutative group is often writte n and referred to as addition, in which case the identity element is written as 0, and the inverse of x as —x. When the group operation is written as multiplication, the opera- tion symbol * is typically written as a dot (i.e., x-y rathe r than x* y) 1
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