• Document: INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL
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SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics and Computer Sciences The Algebra Group Kuwait University 2006 ii Contents Preface v List of symbols vii 1 Matrices 11 3 Determinants 29 4 n-Vectors 37 5 Lines and Planes 41 6 Vector Spaces 45 8 Diagonalization 55 References 59 iii iv CONTENTS Preface Back in 1997, somebody asked in the Mathematics Department: ”Why are the results in 111 Course (Linear Algebra), so bad?” The solution was to cancel some sections of the 6 chapters selected for this one semester course. The solutions of some of the so-called theoretical Exercises were to be covered in the lectures. But this takes time and less time remains for covering the long material in these 6 chapters. Our collection of solutions is intended to help the students in 111 course, and provides to the lecturers a precious additional time in order to cover carefully all the large number of notions, results, examples and procedures to be taught in the lectures. Moreover, this collection yields all the solutions of the Chapter Tests and as a Bonus, some special Exercises to be solved by the students in their Home Work. Because often these Exercises are required in Midterms and Final Exam, the students are warmly encouraged to prepare carefully these solutions, and, if some of them are not understood, to use the Office Hours of their teachers for supplementary explanations. The author v vi PREFACE List of symbols Symbol Description N the set of all positive integer numbers Z the set of all integer numbers Q the set of all rational numbers R the set of all real numbers for R any of the above numerical sets R∗ the set R, removing zero Rn the set of all n-vectors with entries in R Mm×n (R) the set of all m × n matrices with entries in R Mn (R) the set of all (square) n × n matrices Sn the set of all permutations of n elements P(M ) the set of all subsets of M R[X] the set of all polynomials of indeterminate X with coefficients in R vii viii LIST OF SYMBOLS Second Edition (updated to eighth Edition) All rights reserved to the Department of Mathematics and Computer Sciences, Faculty of Science, Kuwait University ix 10 Chapter 1 Matrices Page 20. T.5. A square matrix A = [aij ] is called upper triangular if aij = 0 for i > j. It is called lower triangular if aij = 0 for i < j. a11 a12 ... ... ... a1n    0 a22 ... ... ... a2n     0 0 a33 ... ... a3n   . .. .. . . ..   . .  . . . .    . .. .. .. ..   .. . . . .  0 0 0 ... 0 ann Upper triangular matrix (The entries below the main diagonal are zero.) a11 0 0 ... ... 0     a21 a22 0 ... ... 0    a31 a32 a33 0 ... 0   .. .. .. . . ..    . . . . .    .. .. .. ..   . . . . 0  an1 an2 an3 ... ... ann 11 12 CHAPTER 1. MATRICES Lower triangular matrix (The entries above the main diagonal are zero.) (a) Show that the sum and difference of two upper triangular matrices is upper triangular. (b) Show that the sum and difference of two lower triangular matrices is lower triangular. (c) Show that if a matrix is upper and lower triangular, then it is a diagonal matrix. Solution. (a) As A above, let B = [bij ] be also an upper triangular matrix, S = A+B = [sij ] be the sum and D = A−B = [dij ] be

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