2 Basic Linear Algebra

# 2 Basic Linear Algebra - Basic Linear Algebra In this...

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± ± ± ± ± ± ± ± ± ± ± Basic Linear Algebra In this chapter, we study the topics in linear algebra that will be needed in the rest of the book. We begin by discussing the building blocks of linear algebra: matrices and vectors. Then we use our knowledge of matrices and vectors to develop a systematic procedure (the Gauss– Jordan method) for solving linear equations, which we then use to invert matrices. We close the chapter with an introduction to determinants. The material covered in this chapter will be used in our study of linear and nonlinear programming. 2.1 Matrices and Vectors Matrices DEFINITION A matrix is any rectangular array of numbers. For example, ±² , , ,[ 2 1 ] are all matrices. If a matrix A has m rows and n columns, we call A an m ± n matrix. We refer to m ± n as the order of the matrix. A typical m ± n matrix A may be written as A ² The number in the i th row and j th column of A is called the ij th element of A and is written a ij . For example, if A ² then a 11 ² 1, a 23 ² 6, and a 31 ² 7. 3 6 9 2 5 8 1 4 7 a 1 n a 2 n ³ ³ ³ a mn ³³³ ³³³ ³ ³ ³ ³³³ a 12 a 22 ³ ³ ³ a m 2 a 11 a 21 ³ ³ ³ a m 1 1 ´ 2 3 6 2 5 1 4 2 4 1 3

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Sometimes we will use the notation A ± [ a ij ] to indicate that A is the matrix whose ij th element is a ij . DEFINITION Two matrices A ± [ a ij ] and B ± [ b ij ] are equal if and only if A and B are of the same order and for all i and j , a ij ± b ij . For example, if A ± ±² and B ± then A ± B if and only if x ± 1, y ± 2, w ± 3, and z ± 4. Vectors Any matrix with only one column (that is, any m ² 1 matrix) may be thought of as a column vector. The number of rows in a column vector is the dimension of the column vector. Thus, may be thought of as a 2 ² 1 matrix or a two-dimensional column vector. R m will denote the set of all m -dimensional column vectors. In analogous fashion, we can think of any vector with only one row (a 1 ² n matrix as a row vector. The dimension of a row vector is the number of columns in the vector. Thus, [9 2 3] may be viewed as a 1 ² 3 matrix or a three-dimensional row vector. In this book, vectors appear in boldface type: for instance, vector v. An m -dimensional vector (either row or column) in which all elements equal zero is called a zero vector (written 0 ). Thus, [ 00 ] and are two-dimensional zero vectors. Any m -dimensional vector corresponds to a directed line segment in the m -dimensional plane. For example, in the two-dimensional plane, the vector u ± corresponds to the line segment joining the point to the point The directed line segments corresponding to u ± , v ± , w ± are drawn in Figure 1. ³ 1 ³ 2 1 ³ 3 1 2 1 2 0 0 1 2 0 0 1 2 y z x w 2 4 1 3 12 CHAPTER 2 Basic Linear Algebra
The Scalar Product of Two Vectors An important result of multiplying two vectors is the scalar product. To deﬁne the scalar prod- uct of two vectors, suppose we have a row vector u = [ u 1 u 2 ±±± u n ] and a column vector v ² ±² of the same dimension. The scalar product of u and v (written u ± v ) is the number u 1 v 1 ³ u 2

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