Chapter 1 Preliminaries This chapter is a brief summary of some basic facts, notation, and terminology for vector spaces, linear transformations, and matrices. It is assumed the reader is already familiar with most of the contents of this chapter. 1.1. Vector Spaces A field F is a nonempty set with two binary operations, called addition and multi- plication, satisfying the following: Addition is commutative and associative. There is an additive identity, called zero (0), and every element of F has an additive inverse. Multiplication is commutative and associative. There is a multiplicative iden- tity, called one (1), which is different from 0. Every nonzero element of F has a multiplicative inverse. For all elements a, b, and c of F, the distributive law holds, i.e., a(b + c) = ab + ac. Thus, in the language of abstract algebra, a field F is a nonempty set with two binary operations, addition and multiplication, such that F is an abelian group under addition, the set of nonzero elements of F is an abelian group under multiplication, and the distributive law holds, i.e., a(b + c) = ab + ac for all elements a, b, and c of F. Since the multiplicative and additive identities, 0 and 1, are required to be different, any field has at least two elements. The smallest field is the binary field of two elements, F = {0, 1}, where 1+1 = 0. The most familiar fields are the rational numbers Q, the real numbers R, the complex numbers C, and Zp, the finite field of order p, where p is prime when p = 2 we get the binary field. Most of the time we will be working over the field R or C. A vector space V, over a field F, is a nonempty set V, the objects of which are called vectors, with two operations, vector addition and scalar multiplication, such that the following hold: 1
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