ch07

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Part B LINEAR ALGEBRA. VECTOR CALCULUS Part B consists of Chap. 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Chap. 8 Linear Algebra: Matrix Eigenvalue Problems Chap. 9 Vector Differential Calculus. Grad, Div, Curl Chap. 10 Vector Integral Calculus. Integral Theorems. Hence we have retained the previous subdivision of Part B into four chapters. Chapter 9 is self-contained and completely independent of Chaps. 7 and 8. Thus, Part B consists of two large independent units, namely, Linear Algebra (Chaps. 7, 8) and Vector Calculus (Chaps. 9, 10). Chapter 10 depends on Chap. 9, mainly because of the occurrence of div and curl (defined in Chap. 9) in the Gauss and Stokes theorems in Chap. 10. CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Changes The order of the material in this chapter and its subdivision into sections has been retained, but various local changes have been made to increase the usefulness of this chapter for applications, in particular: 1. The beginning, which was somewhat slow by modern standards, has been streamlined, so that the student will see applications to linear systems of equations much earlier. 2. A reference section (Sec. 7.6) on second- and third-order determinants has been included for easier access from other parts of the book. SECTION 7.1. Matrices, Vectors: Addition and Scalar Multiplication, page 272 Purpose. Explanation of the basic concepts. Explanation of the two basic matrix operations. The latter derive their importance from their use in defining vector spaces, a fact that should perhaps not be mentioned at this early stage. Its systematic discussion follows in Sec. 7.4, where it will fit nicely into the flow of thoughts and ideas. Main Content, Important Concepts Matrix, square matrix, main diagonal Double subscript notation Row vector, column vector, transposition Equality of matrices Matrix addition Scalar multiplication (multiplication of a matrix by a scalar) 145 im07.qxd 9/21/05 12:09 PM Page 145

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Comments on Important Facts One should emphasize that vectors are always included as special cases of matrices and that those two operations have properties [formulas (3), (4)] similar to those of operations for numbers, which is a great practical advantage. SOLUTIONS TO PROBLEM SET 7.1, page 277 2. YZ , , , 4. , same, , undefined 6. Undefined, undefined, , same 8. Undefined, , undefined, undefined 10. 1 _ 5 A, _ 1 10 A. Similar (and more important) instances are the scaling of equations in linear systems, the formation of linear combinations, and the like, as will be shown later. 12. 3, 2, 2 4 and 0, 2, 0. The concept of a main diagonal is restricted to square matrices. 14. No, no, no. Transposition, which relates row and column vectors, will be discussed in the next section. 16. (b). The incidence matrices are as follows, with nodes corresponding to rows and branches to columns, as in Fig. 152.
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ch07 - im07.qxd 12:09 PM Page 145 Part B LINEAR ALGEBRA...

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