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This book contains the basics of linear algebra with an emphasis on nonstandard and neat proofs of known theorems. Many of the theorems of linear algebra obtained mainly during the past 30 years are usually ignored in text-books but are quite accessible for students majoring or minoring in mathematics. These theorems are given with complete proofs. There are about 230 problems with solutions. | PROBLEMS AND THEOREMS IN LINEAR ALGEBRA V. Prasolov ABSTRACT. This book contains the basics of linear algebra with an emphasis on nonstandard and neat proofs of known theorems. Many of the theorems of linear algebra obtained mainly during the past 30 years are usually ignored in text-books but are quite accessible for students majoring or minoring in mathematics. These theorems are given with complete proofs. There are about 230 problems with solutions. Typeset by X S-TeX 1 CONTENTS Preface Main notations and conventions Chapter I. Determinants Historical remarks Leibniz and Seki Kova. Cramer L Hospital Cauchy and Jacobi 1. Basic properties of determinants The Vandermonde determinant and its application. The Cauchy determinant. Continued fractions and the determinant of a tridiagonal matrix. Certain other determinants. Problems 2. Minors and cofactors Binet-Cauchy s formula. Laplace s theorem. Jacobi s theorem on minors of the adjoint matrix. The generalized Sylvester s identity. Chebotarev s theorem on the matrix eij 1 1 where e exp 2ni p . Problems 3. The Schur complement Given A A11 A12 the matrix A An A22 A21A-- 1A12 is A21 A22 called the Schur complement of A11 in A . 3.1. det A det A11 det A A11 . 3.2. Theorem. A B A C B C . Problems 4. Symmetric functions sums xk -px and Bernoulli numbers Determinant relations between k x1 xn Sy X1 Xn X - xXn and Pk x1 xn P x 1 xp. A determinant formula for i1 .ifc n Sn k 1n k 1 n. The Bernoulli numbers and Sn k . 4.4. Theorem. Let u S1 x and v S2 x . Then for k 1 there exist polynomials py and qy such that S2k 1 x u2pk u and S2k x vqy u . Problems Solutions Chapter II. Linear spaces Historical remarks Hamilton and Grassmann 5. The dual space. The orthogonal complement Linear equations and their application to the following theorem 5.4.3. Theorem. If a rectangle with sides a and b is arbitrarily cut into squares with sides x -x then C Q and C Q for all i. ab Typeset by Xv S-TeX 1 2 Problems 6. The kernel null space and the .
