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An Introduction to, LINEAR ALGEBRA, , V. KRISHNAMURTHY, Professor of Mathematics, , ¥ V. P. MAINRA, Assistant Professor of Mathematics, , and, , J. L. ARORA, Assistant Professor of Mathematics, , All at the Birla Institute of Technology and Science,, Pilani, , AS Lad, , AFFILIATED EAST-WEST PRESS PVT LTD, NEW DELHI: MADRAS

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Price Rs 22.50, , No reproduction in any form of this book, in whole or in part, {except for brief quotations in critical articles or reviews),, may be made without written permission of the publishers., , Printed in India at Re&ha Printers,.New Delhi-110020, Published by Affiliate® East-West Press Private Limited, 104 Nirmal Tower, 26 Barakhamba Road, New Delhi-110001

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To Our Parents

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Contents, , PREFACE, 1 SETS AND FUNCTIONS, ll Sets J, , 1.2 Operations on Sets 5, , 1.3 Relations 8, , 1.4 Functions 10, , 1.5 Binary Operations 17, , 1.6 Algebraic Structures 20, 1.7 Operations on Functions 24, VECTORS, , 2.1 Space Coordinates 33, , 2.2 Vectots—Addition and Scalar Multiplication, 2.3 Dot Product of Vectors 48, , 2.4 Applications to Geometry 55, VECTOR SPACES, , 3.1 Vector Spaces 63, , 3.2 Subspaces 69, , 3.3 SpanofaSet 74, , 3.4 More about Subspaces 77, , 3.5 Linear Dependence, Independence 85, 3.6 Dimension and Basis 95, , 4 LINEAR TRANSFORMATIONS, , 4.1 Definition and Examples 107, , 4.2 Range and Kernel ofa Linear Map ‘114, 4.3 Rank and Nullity 118, , 4.4 Inverse of a Linear Transformation 122, , 4.5 Consequences of Rank-Nullity Theorem 127, , 4.6 TheSpaceL(U,V) 129, 4.7 Composition of Linear Maps 133, 48 Operator Equations 138, , 4.9 Applications to the Theory of Ordinary Linear ., , Differential Equations [41, , 37, , 63, , 107

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viii / CONTENTS, , 148, , 5S MATRICES, , 3.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, , 61, 6.2, 63, 6.4, 6.5, 6.6, 67, 68, 69, , Matrix Associated with a Linear Map =: /48, , Linear Map Associated with a Matrix 154, , Linear Operations in Mm n 158, , Matrix Multiplication 163, , Rank and Nullity of a Matrix 174, , Transpose of a Matrix and Spec‘al Types of Matrices 180, Elementary Row Operations 185, , Systems of Linear Equation, = /95, , Matrix Inversion 200, , DETERMINANTS 203, , Definition 203, Fundamental Properties of Determinants 207, Proofs of Theorems 2/2, , Cofactors = 2/6, , Determinant Minors and Rank of a Matrix 224, Product of Determinants 227, , Applications to Linear Equations 230, Eigenvalues, Eigenvectors 233, , Wronskians = -238, , 6.10 Cross Product in V,; 241, , 7A, 72, 13, 14, , MORE MATRIX THEORY 250, , Similarity of Matrices 250, , Inner Product Spaces 258, , Orthogonal and Unitary Matrices 264, Application to Reduction of Quadrics 266, , APPENDIX Ordinary Linear Differential Equations 276, , ANSWERS TO PROBLEMS, , INDEX, , BS