A concise introduction to linear algebra

Géza Schay A Concise Introduction to Linear Algebra Department of Mathematics Géza Schay University of Massachusetts Boston, MA, USA Mathematics Subject Classification (2010) 15Axx Library of Congress[.]

Géza Schay

A Concise Introduction

to Linear Algebra

Department of Mathematics

Géza Schay

University of Massachusetts

Boston, MA, USA

Mathematics Subject Classification (2010): 15Axx

Library of Congress Control Number:

Printed on acid-free paper

© Springer Science +Business Media, LLC 2012

Springer New York Dordrecht Heidelberg London

2012934162

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This book is based on the author’s Introduction to Linear Algebra, published by Jones & Bartlett in 1996.

ISBN 978-0-8176-8324-5

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Index 329 Multilinear function, 223

Multiplicity, 257

n-dimensional Euclidean vector space,

11

Natural permutation, 226

Nonparametric equations for a line, 29

Nonsingular matrix, 93

Nontrivial solution, 59

Norm, 16

Normal equations, 200

Normal vector, 34

Normalization, 212

Null space, 108, 124

Nullity, 133

Oblique coordinates, 26

Ohm’s law, 62

One-to-one mapping, 175

Onto mapping, 175

Order of matrix, 87

Ordered basis, 148

Orthogonal complement, 136

Orthogonal decomposition, 211

Orthogonal diagonalization, 274

Orthogonal matrix, 213

Orthogonal projection, 21

Orthogonal subspaces, 136

Orthogonal vectors, 17

Orthographic projection, 193

Orthonormal, 24

Orthonormal decomposition, 212

Orthonormal set of vectors, 212

Outer product, 75

Overdetermined, 48

Parallelogram law, 2

Parametric equations of a line, 28

Parametric equations of a plane, 32

Partial pivoting, 302

Particular solution, 58

Permutation, 226

Permutation matrix, 97

Pivot, 46, 53

Position vector, 1

Power method for eigenvalues, 306

Principal Axis Theorem, 274

Projection, 21, 200, 212

Projection matrix, 195, 203

Proper subspaces, 106

Pseudoinverse, 209

QR factorization, 216

Quadratic form, 275

Radius vector, 1 Range, 175 Rank, 55, 177 Rank deficient, 56 Reduced echelon matrix, 60, 121 Reduction formulas, 235 Right inverse, 96 Right-handed coordinate system, 11 Rotation matrix, 67

Row rank, 133 Row space, 122 Row vector, 44 Row-equivalent matrices, 47

Scalar multiple of a mapping, 68 Scalar multiple of a matrix, 69 Scalar product, 17

Scalars, 2, 280 Scaled partial pivoting, 302 Schr¨odinger’s equation, 288 Schur’s lemma, 278 Similar matrices, 157 Similarity transformation, 157 Singular matrix, 93

Skew-symmetric matrix, 233 Span, 108

Spectral Theorem, 284 Spectral Theorem for Symmetric Matrices, 274

Square matrix – order of, 87 Standard basis, 118 Standard vectors, 23 Submatrix, 54 Subspace, 105 Sum of mappings, 68 Sum of matrices, 69 Sum of subspaces, 136 Symmetric matrix, 203, 273

Tensor product, 172 Tetrahedron, 14 Three-dimensional Euclidean vector space, 9

Trace, 160 Transformation, 163 Transition matrix, 149 Transpose of a matrix, 94 Transpose of a product, 94 Transpose of the inverse, 94 Transposition, 226

Transversal, 36

330 Index

Triangle inequality, 16

Triangle law, 8

Triple product, 252

Trivial solution, 59

Trivial subspaces, 106

Two-dimensional Euclidean vector

space, 4

Underdetermined, 48

Unit matrix, 82

Unit vector, 16

Unitary matrix, 282

Upper triangular matrix, 56 Vandermonde determinant, 234 Vector product, 245

Vector properties, 5 Vector space, 99 – of linear transformations, 181 Viewplane, 193

Work, 23 Zero matrix, 81