Banchoff Wermer Linear Algebra Throught Geometry

Thomas Banchoff\nJohn Wermer\n\nLinear Algebra\nThrough Geometry\n\nWith 81 Illustrations\n\nSpringer-Verlag\nNew York Heidelberg Berlin Thomas Banchoff\nJohn Wermer\nDepartment of Mathematics\nBrown University\nProvidence, RI 02912\nU.S.A.\n\nEditors\nF. W. Gehring\nDepartment of Mathematics\nUniversity of Michigan\nAnn Arbor, MI 48109\nU.S.A.\n\nP. R. Halmos\nDepartment of Mathematics\nUniversity of Indiana\nBloomington, IN 47405\nU.S.A.\n\nAMS Subject Classification: 15-01\n\nCover: Computer-generated projection of a four-dimensional cube (see Figure 4.5, p. 223), by Thomas Banchoff with the assistance of Gerald Weil.\n\nLibrary of Congress Cataloging in Publication Data\nBanchoff, Thomas.\nLinear algebra through geometry.\n(Undergraduate texts in mathematics)\nBibliography: p.\nIncludes index.\n1. Algebras, Linear. I. Wermer, John. II. Title.\nIII. Series.\nQA184.B36 1983 512'.5 82-19446\n\n© 1983 by Springer-Verlag New York, Inc.\nAll rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A.\n\nTypeset by Computype, Inc., St. Paul, MN.\nPrinted in the United States of America\nPrinted and bound by R. R. Donnelley & Sons, Harrisonburg, VA.\n\n987654321\n\nISBN 0-387-90787-4 Springer-Verlag New York Heidelberg Berlin\nISBN 3-540-90787-4 Springer-Verlag Berlin Heidelberg New York To our wives Lynore and Kerstin Preface\n\nIn this book we lead the student to an understanding of elementary linear algebra by emphasizing the geometric significance of the subject.\n\nOur experience in teaching beginning undergraduates over the years has convinced us that students learn the new ideas of linear algebra best when these ideas are grounded in the familiar geometry of two and three dimensions. Many important notions of linear algebra already occur in these dimensions in a non-trivial way, and a student with a confident grasp of these ideas will encounter little difficulty in extending them to higher dimensions and to more abstract algebraic systems. Moreover, we feel that this geometric approach provides a solid basis for the linear algebra needed in engineering, physics, biology, and chemistry, as well as in economics and statistics.\n\nThe great advantage of beginning with a thorough study of the linear algebra of the plane is that students are introduced quickly to the most important new concepts while they are still on the familiar ground of two-dimensional geometry. In short order, the student sees and uses the notions of dot product, linear transformations, determinants, eigenvalues, and quadratic forms. This is done in Chapters 2.0-2.7.\n\nThen the very same outline is used in Chapters 3.0-3.7 to present the linear algebra of three-dimensional space, so that the former ideas are reinforced while new concepts are being introduced.\n\nIn Chapters 4.0-4.2 we deal with geometry in a space of four dimensions and introduce linear transformations and matrices in four variables.\n\nFinally, in Chapters 5.1-5.3 we study systems of linear equations in n unknowns, and in conjunction with such systems we develop the notions of n-dimensional vector algebra and the ideas of subspace, basis, and dimension. Except in a single chapter, the student need only know basic high school algebra and geometry and introductory trigonometry in order to read this book. The exception is Chapter 2.8, Differential Systems, where we assume a knowledge of elementary calculus.\n\nAcknowledgments\n\nWe would like to thank the many students in our classes over the years whose interest and suggestions have helped in the development of this book. Our special thanks go to Dale Cavanaugh, and to Elaine Haste for the work of typing our manuscript. Contents\n\nPreface \nvii\nAcknowledgments \nviii\n\n1.0 \tVectors in the Line \t1\n\n2.0 \tThe Geometry of Vectors in the Plane \t3\n2.1 \tTransformations of the Plane \t23\n2.2 \tLinear Transformations and Matrices \t29\n2.3 \tSums and Products of Linear Transformations \t39\n2.4 \tInverses and Systems of Equations \t49\n2.5 \tDeterminants \t60\n2.6 \tEigenvalues \t74\n2.7 \tClassification of Conic Sections \t84\n2.8 \tDifferential Systems \t96\n\n3.0 \tVector Geometry in 3-Space \t111\n3.1 \tTransformations of 3-Space \t126\n3.2 \tLinear Transformations and Matrices \t130\n3.3 \tSums and Products of Linear Transformations \t135\n3.4 \tInverses and Systems of Equations \t145\n3.5 \tDeterminants \t163\n3.6 \tEigenvalues \t175\n3.7 \tSymmetric Matrices \t190\n3.8 \tClassification of Quadric Surfaces \t202 4.0 \tVector Geometry in 4-Space \t207\n4.1 \tTransformations of 4-Space \t216\n4.2 \tLinear Transformations and Matrices \t224\n\n5.1 \tHomogeneous Systems of Equations \t228\n5.2 \tSubspace, Linear Dependence, Dimension \t236\n5.3 \tInhomogeneous Systems of Equations \t244\n\nAfterword \t253\nIndex \t255 CHAPTER 1.0\nVectors in the Line\n\nAnalytic geometry begins with the line. Every point on the line has a real number as its coordinate and every real number is the coordinate of exactly one point. A vector in the line is a directed line segment from the origin to a point with coordinate x. We denote this vector by a single capital letter X. The collection of all vectors in the line is denoted by R.\n\nWe add two vectors by adding their coordinates, so if U has coordinate u, then X + U has coordinate x + u. To multiply a vector X by a real number r, we multiply the coordinate by r, so the coordinate of rX is r x. The vector with coordinate zero is denoted by 0. (See Fig. 1.1.)\n\nThe familiar properties of real numbers then lead to corresponding properties for vectors in 1-space. For any vectors X, U, and W and any real numbers r and s we have:\n\nX + U = U + X.\n(X + U) + W = X + (U + W).\nFor all X, 0 + X = X = X + 0.\nFor any X, there is a vector -X such that X + (-X) = 0.\nr(X + U) = rX + rU\n(r + s)X = rX + sX\nr(sX) = (rs)X\n1X = X\n\nWe can define the length of a vector X with coordinate x as the absolute value of x, i.e., the distance from the point labeled x to the origin. We denote this length by |X| and we may write |X| = √x2. (We always understand this symbol to stand for the non-negative square root.) Then 0 is the 2\nLinear Algebra Through Geometry\n0 X U X + U\n(-U)X 0 X 2X\nFigure 1.1\nunique vector of the length 0 and there are just two vectors with length 1, with coordinates 1 and -1.