Problems in higher algebra by d FADDEEV, 1968

A K (DALLEIEEB, H C COMPIHCKHrl CBOPHMK no BBICLUEn AJ1FEBPE 1134ATEAbCTBO “HAYKA” MOCKBA D FADDEEV, I SOMINSKY PROBLEMS IN HIGHER ALGEBRA TRANSLATED FROM THE RUSSIAN by GEORGE YANKOVSKY MIR PUBLISHERS MOSCOW UDC 512.8 (075.8)=20 Revised from the 1968 Russian edition Ha atteiuttiocom ii3b1Ke TO THE READER Mir Publishers would be grateful for your comments on the content, translation and design of this book We would also be pleased to receive any other suggestions you may wish to make Our address is: Mir Publishers, Pervy Rizhsky Pereulok, Moscow, USSR Printed in the Union of Soviet Socialist Republics Contents Introduction Part I PROBLEMS CHAPTER I COMPLEX NUMBERS Operations on Complex Numbers Complex Numbers in Trigonometric Form Equations of Third and Fourth Degree Roots of Unity CHAPTER EVALUATION OF DETERMINANTS Determinants of Second and Third Order Permutations Definition of a Determinant Basic Properties of Determinants Computing Determinants Multiplication of Determinants Miscellaneous Problems CHAPTER SYSTEMS OF LINEAR EQUATIONS Cramer's Theorem Rank of a Matrix Systems of Linear Forms Systems of Linear Equations CHAPTER MATRICES I Operations on Square Matrices Rectangular Matrices Some Inequalities CHAPTER POLYNOMIALS AND RATIONAL FUNCTIONS OF ONE VARIABLE 11 11 13 19 21 25 25 26 27 29 31 51 56 61 61 64 66 68 76 76 83 88 Operations on Polynomials Taylor's Formula Multiple Roots 88 Proof of the Fundamental Theorem of Higher Algebra and Allied Questions 92 Factorization into Linear Factors Factorization into Irreducible Factors in the Field of Reals Relationships Between Coefficients and Roots 93 Euclid's Algorithm 97 The Interpolation Problem and Fractional Rational Functions 100 Rational Roots of Polynomials Reducibility and Irreducibility over the Field of Rationals 103 Bounds of the Roots of a Polynomial 107 Sturm's Theorem 108 111 Theorems on the Distribution of Roots of a Polynomial 10 Approximating Roots of a Polynomial 115 CHAPTER SYMMETRIC FUNCTIONS 116 I Expressing Symmetric Functions in Terms of Elementary Symmetric Functions Computing Symmetric Functions of the Roots 116 of an Algebraic Equation Power Sums 121 123 Transformation of Equations 124 Resultant and Discriminant The Tschirnhausen Transformation and Rationalization of 129 the Denominator Polynomials that Remain Unchanged under Even Permutations of the Variables Polynomials that Remain Unchanged under Cir130 cular Permutations of the Variables 133 CHAPTER LINEAR ALGEBRA Subspaces and Linear Manifolds Transformation of Coordinates 133 135 Elementary Geometry of n-Dimensional Euclidean Space 139 Eigenvalues and Eigenvectors of a Matrix 141 Quadratic Forms and Symmetric Matrices 146 Linear Transformations Jordan Canonical Form PART II HINTS TO SOLUTIONS CHAPTER I COMPLEX NUMBERS 151 CHAPTER EVALUATION OF DETERMINANTS 153 CHAPTER MATRICES 159 CHAPTER POLYNOMIALS AND RATIONAL FUNCTIONS OF ONE VARIABLE 160 CHAPTER SYMMETRIC FUNCTIONS 164 CHAPTER LINEAR ALGEBRA 166 PART III ANSWERS AND SOLUTIONS CHAPTER I COMPLEX NUMBERS 168 CHAPTER EVALUATION OF DETERMINANTS 186 CHAPTER SYSTEMS OF LINEAR EQUATIONS 196 CHAPTER MATRICES 203 CHAPTER POLYNOMIALS AND RATIONAL FUNCTIONS OF ONE VARIABLE 221 CHAPTER SYMMETRIC FUNCTIONS 261 CHAPTER LINEAR ALGEBRA 286 INDEX 313 INTRODUCTION This book of problems in higher algebra grew out of a course of instruction at the Leningrad State University and the Herzen Pedagogical Institute It is designed for students of universities and teacher's colleges as a problem book in higher algebra The problems included here are of two radically different types On the one hand, there are a large number of numerical examples aimed at developing computational skills and illustrating the basic propositions of the theory The authors believe that the number of problems is sufficient to cover work in class, at home and for tests On the other hand, there are a rather large numb:x of problems of medium difficulty and many which will demand all the initiative and ingenuity of the student Many of the problems of this category are accompanied by hints and suggestions to be found in Part I I These problems are starred Answers are given to all problems, some of the problems are supplied with detailed solutions The authors 303 CH LINEAR ALGEBRA + 3x? (k) V2 = x2 x,4 , + V2 4, x V2 , 2, x2 = x i+ x , X3 , xi= — (I) x12 +.1- + 3x'4 2, 1 xi + X2- X3 '2 X4, 1 xi+ x2+ x2-1 41 2, 2-2 x i + V2 x x'= , V 1, x3 + 22 x -,1/ x2= 1 x2= 21 x — 21 X2 + X3- 2X4 = f X? ± X 22 - 3X7, (m) I I x — x2 -2 x3+ x4; , Xi = 7c1+ I 1 X2 +2 X4+ 1 i_ x3 21 xi ' x2 -2 x4 , -= X2 x2= xl= 1 xi— -2 x2+ x2— 2x4 ' 1 x3+ x4; xi— x2— ' x2+ 21 x3+ 21 x4, xi + (n) 5xi2 _ 5x,2 + 3x,2_3x,42, xi _ 1 , 1 x2 x2 2x4 ' x2— xi+ 1 , x2 + 2x3 x4 ' x3— xi— , x4= xi , 952 (a) ill+ I A-'2 21 ,X22 + X32 + (b) 1 2x2— x3+ xl • +x'2); / 11+1 ,2 + • • • + x',f) x1 `x2 +x32 304 PART Ill ANSWERS AND SOLUTIONS where xl= l (xi + xo + + xn); x;=,xiixi +ai2x2+ + ccinx„, i= 2, , n Vn where (ail, ai„ , aid is any orthogonal and normalized fundamental system of solutions of the equation xi +x2 + rc -Fxn =0 + +x`"2 cos n+1 n +1 954 If all the eigenvalues of the matrix A lie in the interval, [a, b], then all the eigenvalues of the matrix A—AE ate negative for A> b and positive for A b and positive for < a Conversely, if the quadratic form (A — XE)X X is negative for A> b and positive for A