This book begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorem for linear maps, including eigenvectors and eigenvalues, quadratic and hermitian forms, diagnolization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. Material in this new edition has been rewritten and reorganized and new exercises have been added.

are immediately verified. This product is positive definite because if X#- 0, then some Xi #- 0, and XiXi > O. Hence o. [V, §2] ORTHOGONAL BASES, POSITIVE DEFINITE CASE Note however that if X = 109 (1, i) then X,X = 1-1 = O. Example 4. Let V be the space of continuous complex-valued functions on the interval [ - n, n]. If I, g E V, we define *
*

one-dimensional vector space over itself. The set of all linear maps of V into K is called the dual space, and will be denoted by V*. Thus by definition V* = ..

ORTHOGONALITY Proof. If W = {O}, the theorem is immediate. Assume W 0/= {O}, and let {w 1 , ••• ,wr } be a basis of W Extend this basis to a basis of V. Let {({>I' .•. '({>n} be the dual basis. We shall now show that {({>r+1' ... '({>n} is a basis of Wl.. Indeed, ({>/W) = 0 if j = r + 1, ... ,n, so {({>r+1' ... '({>n} is a basis of a subspace of Wl.. Conversely, let ({> E Wl.. Write Since ({>(W) = 0 we have ({>(wJ = ai = 0 for i = 1, ... ,r. Hence ({> lies in the space generated by

The first edition of this book appeared under the titIe Introduction to Linear Algebra © 1970 by Addison-Wesley, Reading, MA. The second edition appeared under the titIe Linear Algebra © 1971 by Addison-Wesley, Reading, MA. © 1987 Springer Science+Bussiness Media, Inc. Originally published by Springer Science+Business Media New York in 1987. Softcover reprint ofthe hardcover 3rd edition 1987 All rights reserved. This work may not be translated or copied in whole or in part without the written

independent. (a) 1, t (b) t, t 2 (c) t, t 4 (d) e', t (e) te', e 2 ' (f) sin t, cos t (g) t, sin t (h) sin t, sin 2t (i) cos t, cos 3t 6. Consider the vector space of functions defined for t > O. Show that the following pairs of functons are linearly independent. (a) t, l/t (b) e', log t 7. What are the coordinates of the function 3 sin t to the basis {sin t, cos t}? + 5 cos t = J(t) with respect 8. Let D be the derivative d/dt. Let J(t) be as in Exercise 7. What are the coordinates of the