[Wiley Series in Probability and Statistics] Advanced Calculus with Applications in Statistics || Basic Concepts in Linear Algebra

✍ Scribed by Khuri, André I.

• Publisher: John Wiley & Sons, Inc.
• Year: 2002
• Tongue: English
• Weight: 267 KB
• Category: Article
• ISBN-13: 9780471394884
• DOI: 10.1002/0471394882.ch2

✦ Synopsis

In this chapter we present some fundamental concepts concerning vector spaces and matrix algebra. The purpose of the chapter is to familiarize the reader with these concepts, since they are essential to the understanding of some of the remaining chapters. For this reason, most of the theorems in this chapter will be stated without proofs. There are several excellent books on Ž linear algebra that can be used for a more detailed study of this subject see . the bibliography at the end of this chapter .

In statistics, matrix algebra is used quite extensively, especially in linear Ž . models and multivariate analysis. The books by Basilevsky 1983 , Graybill Ž . Ž . Ž . 1983 , Magnus and Neudecker 1988 , and Searle 1982 include many applications of matrices in these areas.

In this chapter, as well as in the remainder of the book, elements of the set of real numbers, R, are sometimes referred to as scalars. The Cartesian product = n R is denoted by R n , which is also known as the n-dimensional is1 Euclidean space. Unless otherwise stated, all matrix elements are considered to be real numbers.

2.1. VECTOR SPACES AND SUBSPACES

A vector space over R is a set V of elements called vectors together with two operations, addition and scalar multiplication, that satisfy the following conditions:

1. u q v is an element of V for all u, v in V. 2. If ␣ is a scalar and u g V, then ␣ u g V. 3. u q v s v q u for all u, v in V.

Ž

. Ž . 4. u q v q w s u q v q w for all u, v, w in V. 5. There exists an element 0 g V such that 0 q u s u for all u in V. This element is called the zero vector.