Linear Algebra Through Geometry (Undergraduate Texts in Mathematics)
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This book introduces the concepts of linear algebra through the careful study of two and three-dimensional Euclidean geometry. This approach makes it possible to start with vectors, linear transformations, and matrices in the context of familiar plane geometry and to move directly to topics such as dot products, determinants, eigenvalues, and quadratic forms. The later chapters deal with n-dimensional Euclidean space and other finite-dimensional vector space.
and t2, respectively. Then XI and X2 are orthogonal. PROOF. First assume that b =1= O. Set XI = (;\), X2 = (;:). Then (~ ~)(;:) = t l (;:). So or bYI = (tl - a)xl . If XI XI = 0, then bYI = 0, and since b =1= 0 by assumption, then YI =(g), contrary to hypothesis. SO, XI =1= 0 and 2i = tl XI = 0, so a b Similarly, x 2 =1= 0 and Yl x2 t2 - a = -b- SinceYI/xl andYl/x 2 are the slopes of XI and X2 , to prove that XI and X2 are orthogonal amounts to showing that (~II ) (~: ) =
Chapter 3.1 we examined a number of transformations T of 3-space, all of which have the pmperty tha~ in tenns of the coordinates of X = [:: the coordinates of T(X) are given by linear functions of these coordinates. In each case the formulae are of the following type: T[::] = [:::: :+ :::::+ ::::]. CtX t X3 C2X 2 C3X3 Any transformation of this form is called a linear transformation of 3-space. The expression at [h t a2 h2 a3 ] h3 Ct C2 C3 is called the matrix of the transformation
description, we can establish the following algebraic properties of vector addition and scalar multiplication which are analogous to familiar facts about arithmetic of numbers: (4) (5) (6) X+U=U+X. (X + U) + A = X + (U + A). There is a vector 0 such that X + 0 = X = 0 + X for all X. (7) For any X there is a vector - X such that X + ( - X) = O. (8) reX + U) = rX + rU. (9) (r + s)(X) = rX + sX. (10) r(sX) = (rs)X. (II) I . X = X for each X. Commutative law for vectors Associative law for vectors
(Y) = X . A (Y), so A satisfies A (X) . Y = X . A (Y) whenever X', Y lie in 'IT. Each vector X in X where x, = X· F" X2 'IT (6) can be expressed as = x,F, + X2F2, = X· F 2• We identify X with the vector (~~) in 1R2, and in this way 'IT becomes identified with 1R2. Also, since A takes 'IT into itself, A gives rise to a linear transformation AO of 1R2. For each X = x,F, + x 2F 2 in 'IT, A (X) is identified with AO(~~) in 1R2 (see Fig. matrix of AO? Since A(F,) and A (F2) lie in
Includes bibliographical references and index. ISBN-13: 978-1-4612-8752-0 DOl 10.1007/978-1-4612-4390-8 e-ISBN-13: 978-1-4612-4390-8 1. Algebras, Linear. I. Wermer, John. QA184.B36 1991 512'.5-dc20 II. Title. III. Series. 91-18083 Printed on acid-free paper. © 1992 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 2nd edition 1992 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag