Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world

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2 Pearson Education Limited Edinburgh Gate Harlow Esse CM2 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: wwwpearsonedcouk Pearson Education Limited 214 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6 1 Kirby Street, London EC1N 8TS All trademarks used herein are the property of their respective owners The use of any trademark in this tet does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners ISBN 1: ISBN 13: British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America

3 The first two columns of A are the pivot columns and hence form a basis of Col A = W Hence {v 1, v 2 } is a basis for W Note that the reduced echelon form of A is not needed in order to locate the pivot columns 3 Neither v 1 nor v 2 is in H, so{v 1, v 2 } cannot be a basis for H In fact, {v 1, v 2 } is a basis for the plane of all vectors of the form (c 1, c 2, ), buth is only a line 4 COORDINATE SYSTEMS An important reason for specifying a basis B for a vector space V is to impose a coordinate system on V This section will show that if B contains n vectors, then the coordinate system will make V act like R n If V is already R n itself, then B will determine a coordinate system that gives a new view of V The eistence of coordinate systems rests on the following fundamental result THEOREM 7 The Unique Representation Theorem Let B ={b 1,,b n } be a basis for a vector space V Then for each in V, there eists a unique set of scalars c 1,,c n such that = c 1 b 1 + +c n b n (1) PROOF Since B spans V, there eist scalars such that (1) holds Suppose also has the representation = d 1 b 1 + +d n b n for scalars d 1,,d n Then, subtracting, we have = = (c 1 d 1 )b 1 + +(c n d n )b n (2) Since B is linearly independent, the weights in (2) must all be zero That is, c j = d j for 1 j n DEFINITION Suppose B ={b 1,,b n } is a basis for V and is in V Thecoordinates of relative to the basis B (or the B-coordinates of ) are the weights c 1,,c n such that = c 1 b 1 + +c n b n If c 1,,c n are the B-coordinates of, then the vector in R n c 1 [ ] B = is the coordinate vector of (relative to B), ortheb-coordinate vector of The mapping [ ] B is the coordinate mapping (determined by B)1 c n 1 The concept of a coordinate mapping assumes that the basis B is an indeed set whose vectors are listed in some fied preassigned order This property makes the definition of [ ] B unambiguous 232

4 [ ] EXAMPLE 1 Consider a basis B ={b 1, b 2 } for R 2 1, where b 1 = [ ] [ ] 1 b 2 = Suppose an in R has the coordinate vector [ ] B = Find 3 and SOLUTION The B-coordinates of tell how to build from the vectors in B That is, = ( 2)b 1 + 3b 2 = ( 2) + 3 = 2 6 EXAMPLE 2 The entries in the vector = the standard basis E ={e 1, e 2 }, since 1 1 = = 1 e e 2 If E ={e 1, e 2 }, then [ ] E = A Graphical Interpretation of Coordinates [ 1 6 ] are the coordinates of relative to A coordinate system on a set consists of a one-to-one mapping of the points in the set into R n For eample, ordinary graph paper provides a coordinate system for the plane when one selects perpendicular aes and a unit of measurement on each ais Figure 1 shows the standard [ ] basis {e 1, e 2 }, the vectors b 1 (= e 1 ) and b 2 from Eample 1, and the 1 vector = The coordinates 1 and 6 give the location of relative to the standard 6 basis: 1 unit in the e 1 direction and 6 units in the e 2 direction Figure 2 shows the vectors b 1, b 2, and from Fig 1 (Geometrically, the three vectors lie on a vertical line in both figures) However, the standard coordinate grid was erased and replaced by[ a grid ] especially adapted to the basis B in Eample 1 The 2 coordinate vector [ ] B = gives the location of on this new coordinate system: 3 2 units in the b 1 direction and 3 units in the b 2 direction e 2 b 2 b 1 = e 1 b 2 b 1 FIGURE 1 Standard graph paper FIGURE 2 B-graph paper EXAMPLE 3 In crystallography, the description of a crystal lattice is aided by choosing a basis {u, v, w} for R 3 that corresponds to three adjacent edges of one unit cell of the crystal An entire lattice is constructed by stacking together many copies of one cell There are fourteen basic types of unit cells; three are displayed in Fig Adapted from The Science and Engineering of Materials, 4th Ed, by Donald R Askeland (Boston: Prindle, Weber & Schmidt, c 22), p

5 w w w u v u v u v (a) Simple monoclinic FIGURE 3 Eamples of unit cells (b) Body-centered cubic (c) Face-centered orthorhombic The coordinates of atoms within the crystal are given relative to the basis for the lattice For instance, 1/2 1/2 1 identifies the top face-centered atom in the cell in Fig 3(c) Coordinates in R n When a basis B for R n is fied, the B-coordinate vector of a specified is easily found, as in the net eample EXAMPLE 4 Let b 1 = coordinate vector [ ] B of relative to B SOLUTION The B-coordinates c 1, c 2 of satisfy c 1 + c 1 2 = , b 1 2 =, =, and B ={b 1 5 1, b 2 }Findthe b 2 b 1 FIGURE 4 The B-coordinate vector of is (3, 2) or b 1 [ ] b 1 b 2 [ c1 b 2 ] [ ] 4 = c 2 5 This equation can be solved by row operations on an augmented matri or by using the inverse of the matri on the left In any case, the solution is c 1 = 3, c 2 = 2 Thus = 3b 1 + 2b 2, and [ ] [ ] c1 3 [ ] B = = c 2 2 See Fig 4 (3) The matri in (3) changes the B-coordinates of a vector into the standard coordinates for An analogous change of coordinates can be carried out in R n for a basis B ={b 1,,b n }Let P B =[b 1 b 2 b n ] 234

6 Then the vector equation is equivalent to = c 1 b 1 + c 2 b 2 + +c n b n = P B [ ] B (4) We call P B the change-of-coordinates matri from B to the standard basis in R n Left-multiplication by P B transforms the coordinate vector [ ] B into The changeof-coordinates equation (4) is important Since the columns of P B form a basis for R n, P B is invertible (by the Invertible Matri Theorem) Left-multiplication by PB 1 converts into its B-coordinate vector: PB 1 =[ ] B The correspondence [ ] B, produced here by PB 1, is the coordinate mapping mentioned earlier Since PB 1 is an invertible matri, the coordinate mapping is a oneto-one linear transformation from R n onto R n, by the Invertible Matri Theorem This property of the coordinate mapping is also true in a general vector space that has a basis, as we shall see The Coordinate Mapping Choosing a basis B ={b 1,,b n } for a vector space V introduces a coordinate system in V The coordinate mapping [ ] B connects the possibly unfamiliar space V to the familiar space R n See Fig 5 Points in V can now be identified by their new names [ ] B [] B V n FIGURE 5 The coordinate mapping from V onto R n THEOREM 8 Let B ={b 1,,b n } be a basis for a vector space V Then the coordinate mapping [ ] B is a one-to-one linear transformation from V onto R n PROOF Take two typical vectors in V,say, u = c 1 b 1 + +c n b n w = d 1 b 1 + +d n b n Then, using vector operations, u + w = (c 1 + d 1 )b 1 + +(c n + d n )b n 235

7 It follows that [ u + w ] B = c 1 + d 1 c n + d n = c 1 c n + d 1 d n =[u ] B +[w ] B So the coordinate mapping preserves addition If r is any scalar, then So ru = r(c 1 b 1 + +c n b n ) = (rc 1 )b 1 + +(rc n )b n [ ru ] B = rc 1 rc n = r c 1 c n = r[ u ] B Thus the coordinate mapping also preserves scalar multiplication and hence is a linear transformation See Eercises 23 and 24 for verification that the coordinate mapping is one-to-one and maps V onto R n The linearity of the coordinate mapping etends to linear combinations If u 1,,u p are in V and if c 1,,c p are scalars, then [ c 1 u 1 + +c p u p ] B = c 1 [ u 1 ] B + +c p [ u p ] B (5) In words, (5) says that the B-coordinate vector of a linear combination of u 1,,u p is the same linear combination of their coordinate vectors The coordinate mapping in Theorem 8 is an important eample of an isomorphism from V onto R n In general, a one-to-one linear transformation from a vector space V onto a vector space W is called an isomorphism from V onto W (iso from the Greek for the same, and morph from the Greek for form or structure ) The notation and terminology for V and W may differ, but the two spaces are indistinguishable as vector spaces Every vector space calculation in V is accurately reproduced in W, and vice versa In particular, any real vector space with a basis of n vectors is indistinguishable from R n See Eercises 25 and 26 EXAMPLE 5 Let B be the standard basis of the space P 3 of polynomials; that is, let B ={1, t, t 2, t 3 } A typical element p of P 3 has the form p(t) = a + a 1 t + a 2 t 2 + a 3 t 3 Since p is already displayed as a linear combination of the standard basis vectors, we conclude that [ p ] B = Thus the coordinate mapping p [p ] B is an isomorphism from P 3 onto R 4 All vector space operations in P 3 correspond to operations in R 4 If we think of P 3 and R 4 as displays on two computer screens that are connected via the coordinate mapping, then every vector space operation in P 3 on one screen is eactly duplicated by a corresponding vector operation in R 4 on the other screen The vectors on the P 3 screen look different from those on the R 4 screen, but they act as vectors in eactly the same way See Fig 6 a a 1 a 2 a 3 236