Selected Titles in This Series 151 S . Yu . Slavyanov , Asymptotic solutions o f the one-dimensional Schrodinge r equation , 199 6 150 B . Ya. Levin , Lectures on entire functions, 199 6 149 Takash i Sakai, Riemannian geometry , 199 6 148 Vladimi r I. Piterbarg, Asymptotic methods i n the theory o f Gaussian processe s and fields, 1996 147 S . G . Gindiki n an d L. R. Volevich , Mixed problem fo r partia l differentia l equation s with quasihomogeneous principa l part, 199 6 146 L . Ya. Adrianova , Introduction t o linear system s of differential equations , 199 5 145 A . N. Andrianov and V. G. Zhuravlev , Modular form s an d Heck e operators, 199 5 144 O . V. Troshkin, Nontraditional method s in mathematical hydrodynamics , 199 5 143 V . A. Malyshev an d R. A. Minlos, Linear infinite-particl e operators , 199 5 142 N . V . Krylov, Introduction t o the theory o f diffusio n processes , 199 5 141 A . A. Davydov, Qualitative theor y o f control systems , 199 4 140 Aizi k I. Volpert, Vitaly A. Volpert, an d Vladimir A. Volpert, Traveling wav e solutions o f parabolic systems, 199 4 139 I . V . Skrypnik, Methods for analysi s o f nonlinear ellipti c boundary valu e problems, 199 4 138 Yu . P. Razmyslov, Identities o f algebras and their representations, 199 4 137 F . I. Karpelevich and A. Ya. Kreinin , Heavy traffic limit s for multiphas e queues, 199 4 136 Masayosh i Miyanishi, Algebraic geometry, 199 4 135 Masar u Takeuchi, Modern spherica l functions, 199 4 134 V . V. Prasolov, Problems and theorem s in linear algebra, 199 4 133 P . I. Naumkin an d I. A. Shishmarev, Nonlinear nonloca l equations in the theory o f waves, 199 4 132 Hajim e Urakawa, Calculus of variations an d harmonic maps, 199 3 131 V . V. Sharko, Functions on manifolds: Algebrai c and topological aspects, 199 3 130 V . V. Vershinin, Cobordisms and spectral sequences , 199 3 129 Mitsu o Morimoto, An introduction t o Sato' s hyperfunctions, 199 3 128 V . P. Orevkov, Complexity o f proofs an d their transformations i n axiomatic theories, 199 3 127 F . L. Zak, Tangents and secants o f algebraic varieties, 199 3 126 M . L. Agranovskii, Invariant functio n space s o n homogeneous manifold s o f Lie groups and applications, 199 3 125 Masayosh i Nagata, Theory o f commutative fields, 1993 124 Masahis a Adachi, Embeddings an d immersions, 199 3 123 M . A. Akivis and B. A. Rosenfeld, Eli e Cartan (1869-1951) , 199 3 122 Zhan g Guan-Hou, Theory o f entire and meromorphic functions : Deficien t an d asymptoti c value s and singula r directions, 199 3 121 LB . Fesenk o an d S. V . Vostokov, Local fields and thei r extensions: A constructive approach, 199 3 120 Takeyuk i Hid a an d Masuyuki Hitsuda, Gaussian processes , 199 3 119 M . V . Karasev an d V. P. Maslov, Nonlinear Poisso n brackets. Geometry an d quantization, 199 3 118 Kenkich i Iwasawa, Algebraic functions, 199 3 117 Bori s Zilber, Uncountably categorica l theories , 199 3 116 G . M. Fel'dman, Arithmetic of probability distributions , and characterization problem s on abelian groups, 199 3 115 Nikola i V . Ivanov, Subgroups o f Teichmuller modular groups , 199 2 114 Seiz o ltd, Diffusion equations , 199 2 113 Michai l Zhitomirskil , Typical singularities o f differential 1-form s and Pfaffia n equations , 199 2 112 S . A. Lomov, Introduction t o the general theory o f singular perturbations, 199 2 111 Simo n Gindikin, Tube domains and the Cauchy problem, 199 2 110 B . V. Shabat, Introduction t o comple x analysis Part II. Functions o f severa l variables, 199 2 (Continued in the back of this publication) Problems and Theorems in Linear Algebra This page intentionally left blank V.V. Prasoiov Problems and Theorems in Linear Algebra 10.1090/mmono/134 BHKTOP BacHJibeBir a IIpacojiO B 3AHAMH H TEOPEML I JIHHEHHOH AJirEBPb l Translated b y D . A . Leite s fro m a n origina l Russia n manuscrip t Translation edite d b y Simeo n Ivano v 2000 Mathematics Subject Classification. Primar y 15-01 . ABSTRACT. Thi s boo k contain s the basic s o f linear algebr a with a n emphasi s o n nonstandard an d neat proof s o f known theorems. Man y o f the theorems o f linear algebra obtained mainly during th e past thirt y year s ar e usuall y ignore d i n textbooks bu t ar e quit e accessibl e fo r student s majorin g or minorin g i n mathematics . Thes e theorem s ar e give n wit h complet e proofs . Ther e ar e abou t 230 problems wit h solutions . Library o f Congres s Cataloging-in-Publicatio n Dat a Prasolov, V . V . (Vikto r Vasil'evich ) [Zadachi i teoremy linemo i algebry . English ] Problems an d theorem s i n linea r algebra/V . V . Prasolov ; [translate d b y D . A . Leite s fro m a n original Russia n manuscript ; translatio n edite d b y Simeo n Ivanov ] p. cm . — (Translation s o f mathematical monographs , ISS N 0065-9282 ; v. 134 ) Includes bibliographica l references . ISBN 0-8218-0236- 4 1. Algebras , Linear . I . Ivanov , Simeon . II . Title . III . Series . QA184.P7313 199 4 94-1333 2 512/.5—dc20 CI P Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed to the Acquisitions Department, America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-229 4 USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 199 4 b y the America n Mathematica l Society . Al l rights reserved . Reprinted wit h correction s 1996 . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . & Printe d o n recycle d paper . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 1 3 1 2 1 1 1 0 0 9 0 8 CONTENTS Preface x v Main notations and conventions xvi i Chapter I. Determinant s 1 Historical remarks: Leibni z and Sek i Kova. Cramer , L'Hospital, Cauch y and Jacobi 1. Basic properties of determinants 1 The Vandermonde determinant an d its application. Th e Cauchy determinant. Continue d frac - tions and the determinant of a tridiagonal matrix. Certai n other determinants. Problems 2. Minors and cofactors 9 Binet-Cauchy's formula. Laplace' s theorem. Jacobi' s theorem on minors of the adjoint matrix. The generalize d Sylvester' s identity . Chebotarev' s theore m o n th e matri x ||£ u||f~ , wher e e — exp(2ni/p). Problems 3. The Schur complement 1 6 GivenA = ( n 1 2 ),thematrix(^4|^ii ) = A22—A21A7, 1 A\2 is called the Schur complement \A2\ A22J (of A\\ in A). 3.1. de t A = de t A u de t (A\A n). 3.2. Theorem . (A\B) = ((A\C)\(B\C)). Problems 4. Symmetri c functions, sums x\ H f-xf , an d BernoulU numbers 1 9 Determinant relations between ok (x\,..., x n), J * (X\ ,..., x„ ) = x f H hx j and p^ (x\, ..., x„ ) = l l n ^2 x\ ... x n . A determinant formula for S„ (k) = 1 " -\ h (k — 1)". The Bernoulli numbers and S n(k). 4.4. Theorem . Let u = S\(x) andv — S2W. Then for k > 1 there exist polynomials p^ and q^ such that S2k+\ W = u 2pjc(u) and S^M = v 4k(u)- Problems Solutions Chapter II. Linea r spaces 3 5 Historical remarks: Hamilton and Grassmann 5. The dual space. The orthogonal complement 3 7 Linear equations and their application to the following theorem: Vlll CONTENTS 5.4.3. Theorem . If a rectangle with sides a andb is arbitrarily cut into squares with sides x\,... ,x„ then — € Q and — £ Qfor all i. a b Problems 6. Th e kernel (null space) and the image (range) o f an operator. Th e quotient space 6.2.1. Theorem . Ker^* = (ImA) 1- andlmA* = (KQTA^. Fredholm's alternative . Kronecker-Capelli' s theorem . Criteri a fo r solvabilit y o f th e matri x equation C = AXB.