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 Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: Pearson Education Limited 204 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 0 Kirby Street, London ECN 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text 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 0: X ISBN 3: 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 Change of Basis for a General Vector Space Everything we have done so far can easily be generalized to apply to any finitedimensional vector space. We begin by defining coordinate vectors for an n-dimensional vector space. Definition Let V be a vector space and let E ={v, v 2,...,v n } be an ordered basis for V.If v is any element of V,thenv can be written in the form v = c v + c 2 v 2 + +c n v n where c, c 2,...,c n are scalars. Thus, we can associate with each vector v a unique vector c = (c, c 2,...,c n ) T in R n. The vector c defined in this way is called the coordinate vector of v with respect to the ordered basis E and is denoted [v] E.The c i s are called the coordinates of v relative to E. The examples considered so far have all dealt with changing coordinates in R 2. Similar techniques could be used forr n. In the case ofr n, the transition matrices will be n n. EXAMPLE 5 Let E =[v, v 2, v 3 ]=[(,, ) T,(2, 3, 2) T,(, 5, 4) T ] F =[u, u 2, u 3 ]=[(,, 0) T,(, 2, 0) T,(, 2, ) T ] Find the transition matrix from E to F. If x = 3v + 2v 2 v 3 and y = v 3v 2 + 2v 3 find the coordinates of x and y with respect to the ordered basis F. Solution As in Example 4, the transition matrix is given by U V = 3 5 = The coordinate vectors of x and y with respect to the ordered basis F are given by [x] F = 0 2 = and The reader may verify that 3 8 [y] F = 0 3 = u 5u 2 + 3u 3 = 3v + 2v 2 v 3 8u + 2u 2 + 3u 3 = v 3v 2 + 2v 3 56

4 If V is any n-dimensional vector space, it is possible to change from one basis to another by means of an n n transition matrix. We will show that such a transition matrix is necessarily nonsingular. To see how this is done, let E ={w,...,w n } and F ={v,...,v n } be two ordered bases for V. The key step is to express each basis vector w j as a linear combination of the v i s: w = s v + s 2 v 2 + +s n v n w 2 = s 2 v + s 22 v 2 + +s n2 v n. (4) w n = s n v + s 2n v 2 + +s nn v n Let v V.Ifx =[v] E, it follows from (4) that v = x w + x 2 w 2 + +x n w n = s j x j v + s 2 j x j v s nj x j v n Thus, if y =[v] F,then and hence y i = s ij x j y = Sx i =,...,n The matrix S defined by (4) is referred to as the transition matrix. Once S has been determined, it is a simple matter to change coordinate systems. To find the coordinates of v = x w + +x n w n with respect to {v,...,v n }, we need only calculate y = Sx. The transition matrix S corresponding to the change of basis from {w,...,w n } to {v,...,v n } can be characterized by the condition Sx = y if and only if x w + +x n w n = y v + + y n v n (5) Taking y = 0 in (5), we see that Sx = 0 implies that x w + +x n w n = 0 Since the w i s are linearly independent, it follows that x = 0. Thus, the equation Sx = 0 has only the trivial solution and hence the matrix S is nonsingular. The inverse matrix is characterized by the condition S y = x if and only if y v + +y n v n = x w + +x n w n Accordingly, S is the transition matrix used to change basis from {v,...,v n } to {w,...,w n }. EXAMPLE 6 Suppose that in P 3 we want to change from the ordered basis [, x, x 2 ] to the ordered basis [, 2x, 4x 2 2]. Since[, x, x 2 ] is the standard basis for P 3, it is easier to find 57

5 the transition matrix from [, 2x, 4x 2 2] to [, x, x 2 ].Since the transition matrix is = + 0x + 0x 2 2x = 0 + 2x + 0x 2 4x 2 2 = 2 + 0x + 4x S = The inverse of S will be the transition matrix from [, x, x 2 ] to [, 2x, 4x 2 2]: 0 S 2 = Given any p(x) = a + bx + cx 2 in P 3, to find the coordinates of p(x) with respect to [, 2x, 4x 2 2], we simply multiply 0 2 a c 0 0 b 2 = 2 c b 0 0 Thus, 4 p(x) = (a + 2 c) + ( 2 b) 2x + 4 c (4x 2 2) We have seen that each transition matrix is nonsingular. Actually, any nonsingular matrix can be thought of as a transition matrix. If S is an n n nonsingular matrix and {v,...,v n } is an ordered basis for V, then define {w, w 2,...,w n } by (4). To see that the w j s are linearly independent, suppose that It follows from (4) that i= 4 x j w j = 0 4 c s ij x j v j = 0 By the linear independence of the v i s, it follows that s ij x j = 0 i =,...,n or, equivalently, Sx = 0 58

6 Since S is nonsingular, x must equal 0. Therefore, w,...,w n are linearly independent and hence they form a basis for V. The matrix S is the transition matrix corresponding to the change from the ordered basis {w,...,w n } to {v,...,v n }. In many applied problems, it is important to use the right type of basis for the particular application. You may consider a number of applications involving the eigenvalues and eigenvectors associated with an n n matrix A. The key to solving these types of problems is to switch to a basis for R n consisting of eigenvectors of A. SECTION 5 EXERCISES. For each of the following, find the transition matrix corresponding to the change of basis from {u, u 2 } to {e, e 2 }: (a) u = (, ) T, u 2 = (, ) T (b) u = (, 2) T, (c) u = (0, ) T, u 2 = (2, 5) T u 2 = (, 0) T 2. For each of the ordered bases {u, u 2 } in Exercise, find the transition matrix corresponding to the change of basis from {e, e 2 } to {u, u 2 }. 3. Let v = (3, 2) T and v 2 = (4, 3) T. For each ordered basis {u, u 2 } given in Exercise, find the transition matrix from {v, v 2 } to {u, u 2 }. 4. Let E =[(5, 3) T,(3, 2) T ] and let x = (, ) T, y = (, ) T,andz = (0, 7) T. Determine the values of [x] E, [y] E,and[z] E. 5. Let u = (,, ) T, u 2 = (, 2, 2) T, u 3 = (2, 3, 4) T. (a) Find the transition matrix corresponding to the change of basis from {e, e 2, e 3 } to {u, u 2, u 3 }. (b) Find the coordinates of each of the following vectors with respect to {u, u 2, u 3 }: (i) (3, 2, 5) T (ii) (,, 2) T (iii) (2, 3, 2) T 6. Let v = (4, 6, 7) T, v 2 = (0,, ) T, v 3 = (0,, 2) T,andletu, u 2,andu 3 be the vectors given in Exercise 5. (a) Find the transition matrix from {v, v 2, v 3 } to {u, u 2, u 3 }. (b) If x = 2v + 3v 2 4v 3, determine the coordinates of x with respect to {u, u 2, u 3 }. 7. Given v =, v 2 2 = 2, S = find vectors w and w 2 so that S will be the transition matrix from {w, w 2 } to {v, v 2 }. 8. Given v = 2, v 6 2 =, S = find vectors u and u 2 so that S will be the transition matrix from {v, v 2 } to {u, u 2 }. 9. Let [x, ] and [2x, 2x + ] be ordered bases for P 2. (a) Find the transition matrix representing the change in coordinates from [2x, 2x + ] to [x, ]. (b) Find the transition matrix representing the change in coordinates from [x, ] to [2x, 2x + ]. 0. Find the transition matrix representing the change of coordinates on P 3 from the ordered basis [, x, x 2 ] to the ordered basis [, + x, + x + x 2 ]. Let E ={u,...,u n } and F ={v,...,v n } be two ordered bases for R n,andset U = (u,...,u n ), V = (v,...,v n ) Show that the transition matrix from E to F can be determined by calculating the reduced row echelon form of (V U). 59

7 6 Row Space and Column Space If A is an m n matrix, each row of A is an n-tuple of real numbers and hence can be considered as a vector in R n.them vectors corresponding to the rows of A will be referred to as the row vectors of A. Similarly, each column of A can be considered as a vector in R m, and we can associate n column vectors with the matrix A. Definition If A is an m n matrix, the subspace of R n spanned by the row vectors of A is called the row space of A. The subspace of R m spanned by the column vectors of A is called the column space of A. EXAMPLE Let A = The row space of A is the set of all 3-tuples of the form α(, 0, 0) + β(0,, 0) = (α, β, 0) The column space of A is the set of all vectors of the form α + β 0 + γ 0 = α 0 0 β Thus the row space of A is a two-dimensional subspace ofr 3, and the column space of A is R 2. Theorem 6. Two row-equivalent matrices have the same row space. Proof If B is row equivalent to A, thenb can be formed from A by a finite sequence of row operations. Thus, the row vectors of B must be linear combinations of the row vectors of A. Consequently, the row space of B must be a subspace of the row space of A. SinceA is row equivalent to B, by the same reasoning, the row space of A is a subspace of the row space of B. Definition The rank of a matrix A, denoted rank(a), is the dimension of the row space of A. 60