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Springer Books on Elementary Mathematics by Serge Lang MATH! Encounters with High School Students 1985, ISBN 96129-1 The Beauty of Doing Mathematics 1985, ISBN 96149-6 Geometry: A High School Course (with G. Murrow), Second Edition 1988, ISBN 96654-4 Basic Mathematics 1988, ISBN 96787-7 A First Course in Calculus, Fifth Edition 1986, ISBN 96201-8 Calculus of Several Variables, Third Edition 1987, ISBN 96405-3 Introduction to Linear Algebra, Second Edition 1986, ISBN 96205-0 Linear Algebra, Third Edition 1987, ISBN 96412-6 Undergraduate Algebra, Second Edition 1990, ISBN 97279-X Undergraduate Analysis, Second Edition 1997, ISBN 94841-4 Complex Analysis, Fourth Edition 1999, ISBN 98592-1 Real and Functional Analysis, Third Edition 1993, ISBN 94001-4.
Serge Lang Department of Mathematics Yale University New Haven, CT 06520 U.S.A. Editorial Board S. Axler Department of Mathematics Michigan State University East Lansing, MI 48824 U.S.A. Ribet Department of Mathelnatics University of California at Berkeley Berkeley, CA U.S.A.
Gehring Department of Mathematics University of Michigan Ann Arbor. MI 48019 U.S.A. Mathematics Subjects Classifications (2000): 15-01 Library of Congress Cataloging in Publication Data Lang, Serge, 1927- Introduction to linear algebra. (Undergraduate texts in mathematics) Includes index.
Algebras, Linear. QA184.L37 1986 512'.5 85-14758 Printed on acid-free paper. The first edition of this book was published by Addison-Wesley Publishing Company, Inc., in 1970.
© 1970, 1986 by Springer-Verlag New York Inc. 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, 175 Fifth Avenue, New York, New York 10010, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis.
Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Printed in the United States of America (ASC/EB) 987 SPIN 10977149 Springer-Verlag IS a part of Springer Science+ Busmess Media springeronlin e.
Preface This book is meant as a short text in linear algebra for a one-term course. Except for an occasional example or exercise the text is logically independent of calculus, and could be taught early. In practice, I expect it to be used mostly for students who have had two or three terms of calculus. The course could also be given simultaneously with, or im- mediately after, the first course in calculus. I have included some examples concerning vector spaces of functions, but these could be omitted throughout without impairing the under- standing of the rest of the book, for those who wish to concentrate exclusively on euclidean space.
Furthermore, the reader who does not like n = n can always assume that n = 1, 2, or 3 and omit other interpre- tations. However, such a reader should note that using n = n simplifies some formulas, say by making them shorter, and should get used to this as rapidly as possible.
Furthermore, since one does want to cover both the case n = 2 and n = 3 at the very least, using n to denote either number avoids very tedious repetitions. The first chapter is designed to serve several purposes.
First, and most basically, it establishes the fundamental connection between linear algebra and geometric intuition. There are indeed two aspects (at least) to linear algebra: the formal manipulative aspect of computations with matrices, and the geometric interpretation. I do not wish to prejudice one in favor of the other, and I believe that grounding formal manipula- tions in geometric contexts gives a very valuable background for those who use linear algebra. Second, this first chapter gives immediately concrete examples, with coordinates, for linear combinations, perpendicu- larity, and other notions developed later in the book.
In addition to the geometric context, discussion of these notions provides examples for. VI PREFACE subspaces, and also gives a fundamental interpretation for linear equa- tions. Thus the first chapter gives a quick overview of many topics in the book. The content of the first chapter is also the most fundamental part of what is used in calculus courses concerning functions of several variables, which can do a lot of things without the more general ma- trices. If students have covered the material of Chapter I in another course, or if the instructor wishes to emphasize matrices right away, then the first chapter can be skipped, or can be used selectively for examples and motivation.
After this introductory chapter, we start with linear equations, matrices, and Gauss elimination. This chapter emphasizes computational aspects of linear algebra. Then we deal with vector spaces, linear maps and scalar products, and their relations to matrices. This mixes both the computational and theoretical aspects.
Determinants are treated much more briefly than in the first edition, and several proofs are omitted. Students interested in theory can refer to a more complete treatment in theoretical books on linear algebra. I have included a chapter on eigenvalues and eigenvectors. This gives practice for notions studied previously, and leads into material which is used constantly in all parts of mathematics and its applications. I am much indebted to Toby Orloff and Daniel Horn for their useful comments and corrections as they were teaching the course from a pre- liminary version of this book. I thank Allen Altman and Gimli Khazad for lists of corrections.
Contents CHAPTER I Vectors. Definition of Points in Space.
Located Vectors. Scalar Prod uct. The Norm of a Vector. Parametric Lines. 34 CHAPTER II Matrices and Linear Equations 42 § 1. Multiplication of Matrices.
Homogeneous Linear Equations and Elimination. Row Operations and Gauss Elimination.
70 §5 Row Operations and Elementary Matrices. Linear Combinations. 85 CHAPTER III Vector Spaces. Linear Combinations. Convex Sets. Linear Independence. The Rank of a Matrix.
Vll1 CONTENTS CHAPTER IV Linear Mappings. Mappings.
Linear Mappings. The Kernel and Image of a Linear Map. The Rank and Linear Equations Again.
The Matrix Associated with a Linear Map. 150 Appendix: Change of Bases. 154 CHAPTER V Composition and Inverse Mappings. Composition of Linear Maps.
164 CHAPTER VI Scalar Products and Orthogonality. Scalar Products. Orthogonal Bases.
Bilinear Maps and Matrices. 190 CHAPTER VII Determinants 195 § 1. Determinants of Order 2.
3 x 3 and n x n Determinants. The Rank of a Matrix and Subdeterminants. Cramer's Rule. Inverse of a Matrix. Determinants as Area and Volume. 221 CHAPTER VIII Eigenvectors and Eigenvalues. Eigenvectors and Eigenvalues.
The Characteristic Polynomial. Eigenvalues and Eigenvectors of Symmetric Matrices. Diagonalization of a Symmetric Linear Map.
255 Appendix. Complex Numbers. 260 Answers to Exercises. 265 Index. CHAPTER Vectors The concept of a vector is basic for the study of functions of several variables. It provides geometric motivation for everything that follows. Hence the properties of vectors, both algebraic and geometric, will be discussed in full.
One significant feature of all the statements and proofs of this part is that they are neither easier nor harder to prove in 3-space than they are in 2-space. Definition of Points in Space We know that a number can be used to represent a point on a line, once a unit length is selected. A pair of numbers (i.e. A couple of numbers) (x, y) can be used to represent a point in the plane. These can be pictured as follows:.
o x (a) Point on a line Figure 1 y -, (x, y) I I I I x (b) Point in a plane We now observe that a triple of numbers (x, y, z) can be used to represent a point in space, that is 3-dimensional space, or 3-space. We simply introduce one more axis.
Figure 2 illustrates this. 2 x-aXIS VECTORS z-aXIS ' ' ' ' ' '- ' ' '- ' ' Figure 2 I, §I (x,y,z) Instead of using x, y, z we could also use (Xl' X 2, X3).
'The line could be called I-space, and the plane could be called 2-space. Thus we can say that a single number represents a point in I-space. A couple represents a point in 2-space. A triple represents a point in 3- space. Although we cannot draw a picture to go further, there is nothing to prevent us from considering a quadruple of numbers. And decreeing that this is a point in 4-space.
A quintuple would be a point in 5-space, then would come a sextuple, septuple, octuple. We let ourselves be carried away and define a point in n-space to be an n-tuple of numbers if n is a posItIve integer. We shall denote such an n-tuple by a capital letter X, and try to keep small letters for numbers and capital letters for points. We call the numbers Xl'.,xn the coordinates of the point X.
For example, in 3-space, 2 is the first coordinate of the point (2,3, -4), and -4 is its third coordinate. We denote n-space by Rn. Most of our examples will take place when n 2 or n 3. Thus the reader may visualize either of these two cases throughout the book. However, three comments must be made.
First, we have to handle n 2 and n 3, so that in order to a void a lot of repetitions, it is useful to have a notation which covers both these cases simultaneously, even if we often repeat the formulation of certain results separately for both cases. I, § 1 DEFINITION OF POINTS IN SPACE 3 Second, no theorem or formula is simpler by making the assumption that n 2 or 3. Third, the case n 4 does occur in physics.
One classical example of 3-space is of course the space we live in. After we have selected an origin and a coordinate system, we can describe the position of a point (body, particle, etc.) by 3 coordi- nates. Furthermore, as was known long ago, it is convenient to extend this space to a 4-dimensional space, with the fourth coordinate as time, the time origin being selected, say, as the birth of Christ-although this is purely arbitrary (it might be more convenient to select the birth of the solar system, or the birth of the earth as the origin, if we could deter- mine these accurately). Then a point with negative time coordinate is a BC point, and a point with positive time coordinate is an AD point. Don't get the idea that 'time is the fourth dimension ', however.
The above 4-dimensional space is only one possible example. In economics, for instance, one uses a very different space, taking for coordinates, say, the number of dollars expended in an industry. For instance, we could deal with a 7-dimensional space with coordinates corresponding to the following industries: 1. Farm products 7.
Fish We agree that a megabuck per year is the unit of measurement. Then a point (1,000, 800, 550, 300, 700, 200, 900) in this 7-space would mean that the steel industry spent one billion dollars in the given year, and that the chemical industry spent 700 mil- lion dollars in that year. The idea of regarding time as a fourth dimension is an old one.
Already in the Encyclopedie of Diderot, dating back to the eighteenth century, d'Alembert writes in his article on 'dimension': Cette maniere de considerer les quantites de plus de trois dimensions est aussi exacte que l'autre, car les lettres peuvent toujours etre regardees com me representant des nombres rationnels ou non. J'ai dit plus haut qu'il n'etait pas possible de concevoir plus de trois dimensions. Un homme d'esprit de rna connaissance croit qu'on pourrait cependant regarder la duree comme une quatrieme dimension, et que Ie produit temps par la solidite serait en quelque maniere un produit de quatre dimensions; cette idee peut etre contestee, mais elle a, ce me semble, quelque merite, quand ce ne serait que celui de la nouveaute. Encyclopedie, Vol. 4 VECTORS I1 Translated, this means: This way of considering quantItIes having more than three dimensions is just as right as the other, because algebraic letters can always be viewed as representing numbers, whether rational or not.
I said above that it was not possible to conceive more than three dimensions. A clever gentleman with whom I am acquainted believes that nevertheless, one could view duration as a fourth dimension, and that the product time by solidity would be somehow a product of four dimensions. This idea may be chal- lenged, but it has, it seems to me, some merit, were it only that of being new. Observe how d'Alembert refers to a 'clever gentleman' when he appar- ently means himeself. He is being rather careful in proposing what must have been at the time a far out idea, which became more prevalent in the twentieth century.
D' Alembert also visualized clearly higher dimensional spaces as 'prod- ucts' of lower dimensional spaces. For instance, we can view 3-space as putting side by side the first two coordinates (x l' x 2 ) and then the third x 3. Thus we write We use the product sign, which should not be confused with other 'products', like the product of numbers.
The word 'product' is used in two contexts. Similarly, we can write There are other ways of expressing R4 as a product, namely This means that we view separately the first two coordinates (x l' x 2 ) and the last two coordinates (X3' x 4 ). We shall come back to such products later. We shall now define how to add points. If A, B are two points, say in 3-space, and then we define A + B to be the point whose coordinates are Example 2. In the plane, if A (1, 2) and B ( - 3, 5), then A + B ( - 2, 7). I, § 1 DEFINITION OF POINTS IN SPACE 5 In 3-space, if A ( - 1, n, 3) and B (j2, 7, - 2), then A + B (j2 - 1, n + 7, 1).
U sing a neutral n to cover both the cases of 2-space and 3-space, the points would be written and we define A + B to be the point whose coordinates are We observe that the following rules are satisfied: 1. (A + B) + C A + (B + C). A + B B + A. If we let o (0, 0.,0) be the point all of whose coordinates are 0, then O+AA+OA for all A. Let A (a l.,an) and let - A ( - at.,- an). Then A + (-A) O. All these properties are very simple, and are true because they are true for numbers, and addition of n-tuples is defined in terms of addition of their components, which are numbers.
Do not confuse the number ° and the n-tuple (0.,0). We usually denote this n-tuple by 0, and also call it zero, because no diffi- cuI ty can occur in practice. We shall now interpret addition and multiplication by numbers geo- metrically in the plane (you can visualize simultaneously what happens in 3-space). Let A (2,3) and B (-1, 1). Then A + B (1, 4). 6 VECTORS The figure looks like a parallelogram (Fig. (1,4) (2,3) ( -1,1) Figure 3 Example 4.
Let A = (3, 1) and B = (1,2). Then A + B = (4,3). I, § 1 We see again that the geometric representation of our addition looks like a parallelogram (Fig.
A+B Figure 4 The reason why the figure looks like a parallelogram can be given in terms of plane geometry as follows. We obtain B = (1, 2) by starting from the origin 0 = (0, 0), and moving 1 unit to the right and 2 up.
To get A + B, we start from A, and again move 1 unit to the right and 2 up. Thus the line segments between 0 and B, and between A and A + B are the hypotenuses of right triangles whose corresponding legs are of the same length, and parallel.
The above segments are therefore parallel and of the same length, as illustrated in Fig. A+B B LJ Figure 5. I, § 1 DEFINITION OF POINTS IN SPACE 7 Example 5. If A (3, 1) again, then - A ( - 3, - 1). If we plot this point, we see that - A has opposite direction to A.
We may view - A as the reflection of A through the origin. A -A Figure 6 We shall now consider multiplication of A by a number. If c is any number, we define cA to be the point whose coordinates are Example 6. If A (2, -1,5) and c 7, then cA (14, -7,35).
I t is easy to verify the rules: 5. C(A + B) cA + cB. If Cl C 2 are numbers, then and Also note that (-I)A-A. What is the geometric representation of multiplication by a number? Let A (1,2) and c 3.
Then cA (3,6) as in Fig. Multiplication by 3 amounts to stretching A by 3.
Similarly,!A amounts to stretching A by , i.e. Shrinking A to half its size.
In general, if t is a number, t 0, we interpret tA as a point in the same direction as A from the origin, butt times the distance. In fact, we define A. 8 VECTORS I, §1 B to have the same direction if there exists a number c 0 such that A = cB.
We emphasize that this means A and B have the same direction with respect to the origin. For simplicity of language, we omit the words 'with respect to the origin'. Mulitiplication by a negative number reverses the direction.
Thus - 3A would be represented as in Fig. 3A = (3,6) 3A -3A (a) (b) Figure 7 We define A, B (neither of which is zero) to have opposite directions if there is a number c. 10 VECTORS I, 2J This means that B A + (B - A) -+ -+ Let AB and CD be two located vectors. We shall say that they are -+ equivalent if B - A D - C. Every located vector AB is equivalent to -+ one whose beginning point is the origin, because AB is equivalent to I O(B - A). Clearly this is the only located vector whose beginning point -+ is the origin and which is equivalent to AB.
If you visualize the parallelo- gram law in the plane, then it is clear that equivalence of two located vectors can be interpreted geometrically by saying that the lengths of the line segments determined by the pair of points are equal, and that the directions' in which they point are the same. I In the next figures, we have drawn the located vectors O(B - A), -+ I -+ AB, and O(A - B), BA.
AB AB B-A o o A-B Figure 10 Figure 11 -+ Example 1. Let P (1, - 1, 3) and Q (2, 4, 1). Then PQ is equiva- -+ lent to OC, where C Q - P (1, 5, -2). If A (4, -2,5) and B (5, 3, 3), -+ -+ then PQ is equivalent to AB because Q - P B - A (1,5, -2).+ Given a located vector OC whose beginning point is the OrIgIn, we -+ shall say that it is located at the origin. Given any located vector AB, we shall say that it is located at A. A located vector at the origin is entirely determined by its end point.
In view of this, we shall call an n-tuple either a point or a vector, de- pending on the interpretation which we have in mind.+ -+ Two located vectors AB and PQ are said to be parallel if there is a number c =1= 0 such that B - A c(Q - P). They are said to have the. I, 2J LOCATED VECTORS 11 same direction if there is a number c ° such that B - A = c(Q - P), and have opposite direction if there is a number c 0, -. We even see that PQ and AB have the same direction. In a similar manner, any definition made concerning n-tuples can be carried over to located vectors. For instance, in the next section, we shall define what it means for n-tuples to be perpendicular.
B Q Q-P:/ B-A o Figure 13. 12 VECTORS I, §3 -+- -+- Then we can say that two located vectors AB and PQ are perpendicular if B - A is perpendicular to Q - P. 13, we have drawn a picture of such vectors in the plane. Exercises I, §2 -+ -+ In each case, determine which located vectors PQ and AB are equivalent. P = (1, -1), Q = (4, 3), A = (-1, 5), B = (5, 2). P = (1,4), Q = (-3,5), A = (5,7), B = (1, 8). P = (1, -1,5), Q = (-2,3, -4), A = (3,1,1), B = (0, 5,10).
P = (2, 3, - 4), Q = ( - 1, 3, 5), A = ( - 2, 3, - 1), B = ( - 5, 3, 8).+ -+ In each case, determine which located vectors PQ and AB are parallel. P = (1, -1), Q = (4, 3), A = (-1, 5), B = (7, 1). P = (1,4), Q = (-3,5), A = (5,7), B = (9,6). P = (1, -1, 5), Q = (- 2, 3, -4), A = (3, 1, 1), B = ( - 3,9, -17).
P = (2,3, -4), Q = (-1,3,5), A = (-2,3, -1), B = (-11,3, -28). Draw the located vectors of Exercises 1, 2, 5, and 6 on a sheet of paper to illustrate these exercises. Also draw the located vectors QP and BA.
Draw the points Q - P, B - A, P - Q, and A-B. Scalar Product It is understood that throughout a discussion we select vectors always in the same n-dimensional space. You may think of the cases n 2 and n 3 only.
In 2-space, let A (aI' a2) and B (b l, b2). We define their scalar product to be In 3-space, let A (aI' a2, a3) and B (b l, b2, b3). We define their scalar product to be In n-space, covering both cases with one notation, let A (aI'.,an) and B (b l,.,b n) be two vectors. We define their scalar or dot product AB to be. I, §3 SCALAR PRODUCT 13 This product is a number. For instance, if A (1, 3, - 2) and B ( - 1, 4, - 3), then A.
B - 1 + 12 + 6 17. For the moment, we do not give a geometric interpretation to this scalar product.
We shall do this later. We derive first some important proper- ties. The basic ones are: SP t. We have A B B A. If A, B, C are three vectors, then A.
(B + C) A. C (B + C) A. If x is a number, then (xA)B x(AB) and A (xB) x(A. If A 0 is the zero vector, then A A 0, and otherwise AA O.
We shall now prove these properties. Concerning the first, we have because for any two numbers a, b, we have ab ba. This proves the first property. For SP 2, let C (c1.,cn). Then and Reordering the terms yields. 14 VECTORS I, 3J which is none other than A B + A.
This proves what we wanted. We leave property SP 3 as an exercise. Finally, for SP 4, we observe that if one coordinate ai of A is not eq ual to 0, then there is a term af # ° and af ° in the scalar prod uct A. Since every term IS 0, it follows that the sum IS 0, as was to be shown. In much of the work which we shall do concerning vectors, we shall use only the ordinary properties of addition, multiplication by numbers, and the four properties of the scalar product. We shall give a formal discussion of these later.
For the moment, observe that there are other objects with which you are familiar and which can be added, subtracted, and multiplied by numbers, for instance the continuous functions on an interval a, bJ (cf. Example 2 of Chapter VI, §1). Instead of writing A A for the scalar product of a vector with itself, it will be convenient to write also A 2.
(This is the only instance when we allow ourselves such a notation. Thus A 3 has no meaning.) As an exer- cise, verify the following identities: A dot product A B may very well be equal to ° without either A or B being the zero vector. For instance, let A = (1,2,3) and B = (2, 1, -). Then AB = ° We define two vectors A, B to be perpendicular (or as we shall also say, orthogonal), if A B = 0. For the moment, it is not clear that in the plane, this definition coincides with our intuitive geometric notion of perpendicularity. We shall convince you that it does in the next section. Here we merely note an example.
Say in R 3, let E1 = (1,0,0), E2 = (0, 1,0), E3 = (0,0,1) be the three unit vectors, as shown on the diagram (Fig. I, §4 THE NORM OF A VECTOR 15 z -t- Y x Figure 14 Then we see that E 1. E2 0, and similarly Ei.
Ej ° if i =1= j. And these vectors look perpendicular. If A (a l' a2' a3), then we observe that the i-th component of A, namely is the dot product of A with the i-th unit vector. We see that A is perpendlcular to Ei (according to our definition of perpendicularity with the dot product) if and only if its i-th component is equal to 0. Exercises I, §3 1. Find A A for each of the following n-tuples. (a) A =(2, -1), B=(-I, 1) (b) A =(-1,3), B=(0,4) (c) A =(2, -1,5), B=(-I, 1, 1) (d) A =(-1, -2,3), B=(-1,3, -4) (e) A = (n, 3, -1), B = (2n, -3,7) (f) A = (15, -2,4), B = (n, 3, -1) 2.
Find A B for each of the above n-tuples. Using only the four properties of the scalar product, verify in detail the identi- ties given in the text for (A + B)2 and (A - B)2.
Which of the following pairs of vectors are perpendicular? (a) (1, -1,1) and (2,1,5) (b) (1, -1,1) and (2,3,1) (c) (-5,2,7) and (3, -1,2) (d) (n,2, 1) and (2, -n,O) 5. Let A be a vector perpendicular to every vector X. Show that A = o. The Norm of a Vector We define the norm of a vector A, and denote by /lAII, the number IIAII.