chapter 3, Matrix Methods and Linear Vector Spaces

Excerpt

Because matrix methods play such an important role in solving systems of linear equations, we devote this chapter to a review of the basic concepts associated with matrix operations. In addition, we also provide a brief treatment of the notion of linear vector space, which arises again in later chapters. The linear vector space concept provides a unifying approach to several common areas of applied mathematics.

3.1 Introduction

The development of engineering mathematics during the past few decades has been greatly affected by the increasing role of linear analysis and extensive use of computers to solve engineering problems. The subject matter of general linear analysis is vast, including differential equations, matrices, linear algebra, vector and tensor analysis, integral equations, and linear vector spaces. In fact, virtually all of the topics covered in this text can be included under the general title of linear analysis; however, in this chapter we concentrate only on matrix methods and linear vector spaces, the latter including some discussion of Hilbert spaces. Like most chapters throughout this text, our treatment is not intended to be exhaustive.

Linear algebra is basically the theory and application of vectors and matrices in connection with the solution of linear transformations, linear systems of differential equations, eigenvalue problems, and so on. Such systems of equations may arise, for example, from the analysis of electrical networks and various frameworks in mechanics such as describing the motion of a system of particles. Matrices are useful in that they permit us to consider a rectangular array of numbers as a single entity; hence, we can perform various operations in a systematic and compact fashion.

3.2 Basic Matrix Concepts and Operations

The solution of systems of linear equations can be greatly systematized by the use of matrix methods, particularly when the number of equations is greater than two. We will denote a matrix here by a bold letter such as A, B, X, Y, x, y, ….

▶ Matrix: A matrix is a rectangular array of numbers, or elements, arranged in m rows and n columns; that is,

We call A an m×n matrix with elements a_ij (i = 1,2,…,m;j = 1,2,…,n). The first subscript denotes the row, while the second subscript identifies the column. It is also customary to write simply A = [a_ij]. If m = n, we say that A is a square matrix.