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LINEAR ALGEBRA

KENNETH HOFFMAN

Professor of Mathematics

Massachusetts Institute of Technology

RAY KUNZE

Professor of Mathematics University of California, Irvine

Second Edition

PRENTICE-HALL, INC. , Englewood Cliffs, New Jersey

Preface

Our original purpose in writing this book was to provide a text for the under graduate linear algebra course at the Massachusetts Institute of Technology. This course was designed for mathematics majors at the junior level, although three fourths of the students were drawn from other scientific and technological disciplines and ranged from freshmen through graduate students. This description of the M.LT. audience for the text remains generally accurate today. The ten years since the first edition have seen the proliferation of linear algebra courses throughout the country and have afforded one of the authors the opportunity to teach the basic material to a variety of groups at Brandeis University, Washington Univer sity (St. Louis), and the University of California (Irvine). Our principal aim in revising Linear Algebra has been to increase the variety of courses which can easily be taught from it. On one hand, we have structured the chapters, especially the more difficult ones, so that there are several natural stop ping points along the way, allowing the instructor in a one-quarter or one-semester course to exercise a considerable amount of choice in the subject matter. On the other hand, we have increased the amount of material in the text, so that it can be used for a rather comprehensive one-year course in linear algebra and even as a reference book for mathematicians. The major changes have been in our treatments of canonical forms and inner product spaces. In Chapter 6 we no longer begin with the general spatial theory which underlies the theory of canonical forms. We first handle characteristic values in relation to triangulation and diagonalization theorems and then build our way up to the general theory. We have split Chapter 8 so that the basic material on inner product spaces and unitary diagonalization is followed by a Chapter 9 which treats sesqui-linear forms and the more sophisticated properties of normal opera tors, including normal operators on real inner product spaces. We have also made a number of small changes and improvements from the first edition. But the basic philosophy behind the text is unchanged. We have made no particular concession to the fact that the majority of the students may not be primarily interested in mathematics. For we believe a mathe matics course should not give science, engineering, or social science students a hodgepodge of techniques, but should provide them with an understanding of basic mathematical concepts.

iii

iv Preface

On the other hand, we have been keenly aware of the wide range of back grounds which the students may possess and, in particular, of the fact that the students have had very little experience with abstract mathematical reasoning. For this reason, we have avoided the introduction of too many abstract ideas at the very beginning of the book. In addition, we have included an Appendix which presents such basic ideas as set, function, and equivalence relation. We have found it most profitable not to dwell on these ideas independently, but to advise the students to read the Appendix when these ideas arise. Throughout the book we have included a great variety of examples of the important conccpts which occur. The study of such examples is of fundamental importance and tends to minimize the number of students who can repeat defini tion, theorem, proof in logical order without grasping the meaning of the abstract concepts. The book also contains a wide variety of graded exercises (about six hundred), ranging from routine applications to ones which will extend the very best students. These exercises are intended to be an important part of the text. Chapter 1 deals with systems of linear equations and their solution by means of elementary row operations on matrices. It has been our practice to spend about six lectures on this material. It provides the student with some picture of the origins of linear algebra and with the computational technique necessary to under stand examples of the more abstract ideas occurring in the later chapters. Chap ter 2 deals with vector spaces, subspaces, bases, and dimension. Chapter 3 treats linear transformations, their algebra, their representation by matrices, as well as isomorphism, linear functionals, and dual spaces. Chapter 4 defines the algebra of polynomials over a field, the ideals in that algebra, and the prime factorization of a polynomial. It also deals with roots, Taylor's formula, and the Lagrange inter polation formula. Chapter 5 develops determinants of square matrices, the deter minant being viewed as an alternating n-linear function of the rows of a matrix, and then proceeds to multilinear functions on modules as well as the Grassman ring. The material on modules places the concept of determinant in a wider and more comprehensive setting than is usually found in elementary textbooks. Chapters^6 and 7 contain a discussion of the concepts which are basic to the analysis of a single linear transformation on a finite-dimensional vector space; the analysis of charac teristic (eigen) values, triangulable and diagonalizable transformations ; the con cepts of the diagonalizable and nilpotent parts of a more general transformation, and the rational and Jordan canonical forms. The primary and cyclic decomposition theorems play a central role, the latter being arrived at through the study of admissible subspaces. Chapter 7 includes a discussion of matrices over a polynomial domain, the computation of invariant factors and elementary divisors of a matrix, and the development of the Smith canonical form. The chapter ends with a dis cussion of semi-simple operators, to round out the analysis of a single operator. Chapter 8 treats finite-dimensional inner product spaces in some detail. It covers the basic geometry, relating orthogonalization to the idea of 'best approximation to a vector' and leading to the concepts of the orthogonal projection of a vector onto a subspace and the orthogonal complement of a subspace. The chapter treats unitary operators and culminates in the diagonalization of self-adjoint and normal operators. Chapter 9 introduces sesqui-linear forms, relates them to positive and self-adjoint operators on an inner product space, moves on to the spectral theory of normal operators and then to more sophisticated results concerning normal operators on real or complex inner product spaces. Chapter 10 discusses bilinear forms, emphasizing canonical forms for symmetric and skew-symmetric forms, as well as groups preserving non-degenerate forms, especially the orthogonal, unitary, pseudo-orthogonal and Lorentz groups. ' We feel that any course which uses this text should cover Chapters 1, 2, and^3

Contents

Contents

Chapter 4. Polynomials

4.1. Algebras

117 4.2. The Algebra of Polynomials 4.3. Lagrange Interpolation 4.4. Polynomial Ideals 4.5. The Prime Factorization of a Polynomial

119 124 127 134

Chapter 5. Determinants 140

Chapter 6.

Chapter 7.

Chapter 8.

5.1. Commutative Rings 140 5.2. Determinant Functions 141 5.3. Permutations and the Uniqueness of Determinants 150 5.4. Additional Properties of Determinants 156 5.5. Modules 164 5.6. Multilinear Functions 166 5.7. The Grassman Ring 173

Elementary Canonical Forms 181

6.1. Introduction 181 6.2. Characteristic Values 182 6.3. Annihilating Polynomials 190 6.4. Invariant Subspaces 198 6.5. Simultaneous Triangulation; Simultaneous Diagonalization 206 6.6. Direct-Sum Decompositions 209 6.7. Invariant Direct Sums 213 6.8. The Primary Decomposition Theorem 219

The Rational and Jordan Forms 227

7.1. Cyclic Subspaces and Annihilators 227 7.2. Cyclic Decompositions and the Rational Form 231 7.3. The Jordan Form 244 7.4. Computation of Invariant Factors 251 7.5. Summary; Semi-Simple Operators 262

Inner Product Spaces 270

8.1. (^) Inner Products 270 8.2. Inner Product Spaces 277 8.3. Linear Functionals and Adjoints 290 8.4. Unitary Operators 299 8.5. Normal Operators 3 1 1

vii

1. Linear Equations

1 .1. Fields

We assume that the reader is familiar with the elementary algebra of real and complex numbers. For a large portion of this book the algebraic properties of numbers which we shall use are easily deduced from the following brief list of properties of addition and multiplication. We let F denote either the set of real numbers or the set of complex numbers.

1. Addition is commutative,

x+y y+x for all x and y in F.

2. Addition is associative,

x+(y+z) =^ (x + y)+z for all x, y, and z in F.

3. There is a unique element 0 (zero) in F^ such that x+ 0 = x, for every x in^ F.
4. To each x in F^ there corresponds a unique element (-x) in F^ such that x+(-x) = O.
5. Multiplication is commutative,

xy =^ yx for all x and y in F.

6. Multiplication is associative,

x(yz) = (xy)z for all x, y, and z in F.

1

2 Linear Equations^ Chap.^1

7. There is a unique non-zero element 1 (one) in F such that x l^ =^ x, for every x in F.

8. To each nOll-zero x in F there corresponds a unique element X-I

(or 1/x) in F such that xx-1 = 1.

9. Multiplication distributes over addition; that is, x(y + z) = xy (^) + xz, for all x, y, and z in F. Suppose one has a set F of objects x, y, z,... and two operations on the elements of F as follows. The first operation, called addition, asso ciates with each pair of elements x, y in F an element (x + y) in F ; the second operation, called multiplication, associates with each pair x, y an element xy in F ; and these two operations satisfy conditions (1)- (9) above. The set F, together with these two operations, is then called a field. Roughly speaking, a field is a set together with some operations on the objects in that set which behave like ordinary addition, subtraction, multiplication, and division of numbers in the sense that they obey the nine rules of algebra listed above. With the usual operations of addition

and multiplication, the set C of complex numbers is a field, as is the set R

of real numbers. For most of this book the 'numbers' we use may as well be the ele ments from any field F. To allow for this generality, we shall use the word 'scalar' rather than 'number.' Not much will be lost to the reader if he always assumes that the field of scalars is a subfield of the field of complex numbers. A subfield of the field C is a set F of complex numbers which is itself a field under the usual operations of addition and multi plication of complex numbers. This means that ° and 1 are in the set F, and that if x and y are elements of F, so are (x + y), -x, xy, and X-I

(if x^ �^ 0). An example of such a subfield is the field^ R^ of real numbers;

for, if we identify the real numbers with the complex numbers (a+ib) for which b = 0, the ° and 1 of the complex field are real numbers, and if x and y are real, so are (x+y), -x, xy, and X-I^ (if x � 0). We shall give other examples below. The point of our discussing subfields is essen tially this : If we are working with scalars from a certain subfield of C, then the performance of the operations of addition, subtraction, multi plication, or division on these scalars does not take us out of the given subfield.

EXAMPLE 1. The set of positive integers: 1, 2, 3,... , is not a sub field of C, for a variety of reasons. For example, ° is not a positive integer; for no positive integer n is - n a positive integer; for no positive integer n except 1 is lin a positive integer.

EXAMPLE 2. The set of integers:... , 2, -1, 0, 1 , 2,... , is not a sub field of C, because for an integer n, lin is not an integer unless n is 1 or

4 Linear Equations^ Chap. 1

equations in (1-1) is called a solution of the system. If (^) YI = (^) Y2 =.^.. Ym 0, we say that the system is^ homogeneous,^ or that each of the equations is homogeneous. Perhaps the most fundamental technique for finding the solutions of a system of linear equations is the technique of elimination. We can illustrate this technique on the homogeneous system 2XI - X2+ Xa = 0 Xl+3X2 (^) + 4xa = O. If we add ( - 2) times the second equation to the first equation, we obtain -7X2 - txa =^0 or, X2 -X3. If we add 3 times the first equation to the second equation, we obtain

7XI+txa = 0

or, Xl = -Xa. So we conclude that if (Xl, X2, Xa) is a solution then Xl =^ X2^ = -Xa. Conversely, one can readily verify that any such triple is a solution. Thus the set of solutions consists of all triples ( a, a, a). We found the solutions to this system of equations by 'eliminating unknowns,' that is, by multiplying equations by scalars and then adding to produce equations in which some of the Xj were not present. We wish to formalize this process slightly so that we may understand why it works, and so that we may carry out the computations necessary to solve a system in an organized manner. For the general system (1-1), suppose we select m scalars CI,. • • , Cm, multiply the jth equation by Cj and then add. We obtain the equation (ClAn+...+CmAml)Xl +... + (clAln +...^ + cmAmn)xn = (^) CIYI+.^.^. + CmYm. Such an equation we shall call a linear combination of the equations in (1-1 ). Evidently, any solution of the entire system of equations (1-1) will also be a solution of this new equation. This is the fundamental idea of the elimination process. If we have another system of linear equations

BnXl + ...+^ B1nxn =^ Zl

Bklxl +... + Bknxn Zk in which each of the k equations is a linear combination of the equations in (1-1), then every solution of (1-1) is a solution of this new system. Of course it may happen that some solutions of (1-2) are not solutions of (1-1 ). This clearly does not happen if each equation in the original system is a linear combination of the equations in the new system. Let us say that two systems of linear equations are equivalent if each equation in each system is a linear combination of the equations in the other system. We can then formally state our observations as follows.

Sec. 1.2 Systems of Linear Equations

Theorem 1. Equivalent systems of linear equations have exactly the

same solutions.

If the elimination process is to be effective in finding the solutions of a system like (1-1), then one must see how, by forming linear combina tions of the given equations, to produce an equivalent system of equations which is easier to solve. In the next section we shall discuss one method of doing this.

Exercises

1. Verify that the set of complex numbers described in Example 4 is a sub field of C.
2. Let F be the field of complex numbers. Are the following two systems of linear equations equivalent? If so, express each equation in each system as a linear combination of the equations in the other system. 3Xl + X2 =^0 Xl + X2 0
3. Test the following systems of equations as in Exercise 2.

• Xl + X2 + 4xa 0 Xl + 3X2 + 8xa^ =^0 �Xl +^ X2 + fXa = 0

Xl (^) - Xa 0 X2 + 3xa =^0

4. Test the following systems as in Exercise 2.

( 1 + �) Xl + 8X2 - iXa - X4 = 0

5. Let F be a set which contains exactly two elements, 0 and 1. Define an addition and multiplication by the tables :

• 0 1 o 0 1 1 1 0

o 1 o 0 0 1 0 1 Verify that the set F, together with these two operations, is a field.

6. Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent.
7. Prove that each subfield of the field of complex numbers contains every rational number.
8. Prove that each field of characteristic zero contains a copy of the rational number field.

5

Sec. 1 .3 Matrices and Elementary Row Operations

In defining e(A ) , it is not really important how many columns A has, but

the number of rows of A is crucial. For example, one must worry a little

to decide what is meant by interchanging rows 5 and 6 of a 5 X 5 matrix. To avoid any such complications, we shall agree that an elementary row

operation e is defined on the class of all m X n matrices over F, for some

fixed m but any n. In other words, a particular e is defined on the class of

all m-rowed matrices over F'. One reason that we restrict ourselves to these three simple types of

row operations is that, having performed such an operation e on a matrix

A , we can recapture A by performing a similar operation on e(A ).

J'heorem 2. 1'0 each elementary row operation e there corresponds an

elementary row operation e1, of the same type as e, such that el(e(A» =

e(e1(A»^ A^ for each^ A.^ In other words, the inverse operation (junction) of

an elementary row operation exists and is an elementary row operation of the

same type.

Proof. (1) Suppose e is the operation which multiplies the rth row

of a matrix by the non-zero scalar c. Let e1 be the operation which multi

plies row r by c-1• (2) Suppose e is the operation which replaces row r by

row r^ plus^ c^ times row^ s, r^ �^ s.^ Let^ el^ be the operation which replaces row^ r

by row r plus ( c) times row s. (3) If e interchanges rows r and s, let el e.

In each of these three cases we clearly have el(e(A » e (e1(A» = A for

each A. I

Definition. If A and B are m X n^ matrices over the field F, we say that

B is row-equivalent to A if B can be obtained from A by a finite sequence

of elementary row operations.

Using Theorem 2, the reader should find it easy to verify the following.

Each matrix is row-equivalent to itself ; if B is ro>v-equivalent to A , then A

is row-equivalent to B ; if B is row-equivalent to A and C is row-equivalent

to B, then C is row-equivalent to A. In other words, row-equivalence is

an equivalence relation (see Appendix).

Theorem 3. If A and B are row-equivalent m X n matrices, the homo

geneous systems of linear equations AX = 0 and B X = 0 have exactly the

same solutions.

Proof. Suppose we pass from A to B by a finite sequence of

elementary row operations :

A Ao -+ A1-+ ... -+ Ak B.

It is enough to prove that the systems AjX = 0 and Ai+1X = 0 have the

same solutions, i.e., that one elementary row operation does not disturb the set of solutions.

7

8 Linear Equations^ Chap.^1

So suppose that B is obtained from A by a single elementary row operation. No matter which of the three types the operation is, (1), (2) , o r (3) , each equation i n the system B X 0 will b e a linear combination of the equations in the system AX = O. Since the inverse of an elementary row operation is an elementary row operation, each equation in AX = 0 will also be a linear combination of the equations in BX O. Hence these two systems are equivalent, and by Theorem 1 they have the same

solutions. I

EXAMPLE 5. Suppose F is the field of rational numbers, and

A

�D^

: J -U

We shall perform a finite sequence of elementary row operations on A, indicating by numbers in parentheses the type of operation performed.

D

[!^

-9^3

[!

1 2"^1

[!

I 2"^1

- n

� [!

n [!

I;J�[�

-EJ^ [!

[!

-9^3

-�J�

=!J

1 2"^1

-l�J�

-:r (^0) -

1 2"^1 -i

-¥J

0 0 -V-

1 1 2" -i

_,l

J

-a-^17

• i The row-equivalence of A with the final matrix i n the above sequence tells us in particular that the solutions of

and

2Xl X2 + 3xa (^) + 2X4 0 Xl + 4X2 - X4^ =^0 2Xl + 6X2 - Xa^ + 5X4^0

Xa - J.";-X4 = 0 Xl + ¥X4 = 0 X2 iX4 = 0 are exactly the same. In the second system it is apparent that if we assign

10 Linear Equations Chap. 1

The second matrix fails to satisfy condition (a) , because the leading non zero entry of the first row is not 1. The first matrix does satisfy condition (a), but fails to satisfy condition (b) in column 3. We shall now prove that we can pass from any given matrix to a row reduced matrix, by means of a finite number of elementary row oper tions. In combination with Theorem 3, this will provide us with an effec tive tool for solving systems of linear equations.

TheQrem 4. Every m X n matrix over the field F is row-equivalent to

a row-reduced matrix.

Proof. Let A be an m X n matrix over F. If every entry in the

first row of A is 0, then condition (a) is satisfied in so far as row 1 is con cerned. If row 1 has a non-zero entry, let k be the smallest positive integer j for which Ali � 0. Multiply row 1 by Ali/, and then condition (a) is

satisfied with regard to row 1. Now for each i 2:: 2, add (-Aik) times row

1 to row �'. Now the leading non-zero entry of row 1 occurs in column le, that entry is 1 , and every other entry in column le is 0. Now consider the matrix which has resulted from above. If every entry in row 2 is 0, we do nothing to row 2. If some entry in row 2 is dif ferent from 0, we multiply row 2 by a scalar so that the leading non-zero entry is 1. In the event that row 1 had a leading non-zero entry in column k, this leading non-zero entry of row 2 cannot occur in column k ; say it

occurs in column kT � k. By adding suitable multiples of row 2 to the

various rows, we can arrange that all entries in column k' are 0, except the 1 in row 2. The important thing to notice is this : In carrying out these last operations, we will not change the entries of row 1 in columns 1,... , k ; nor will we change any entry of column k. Of course, if row 1 was iden

tically 0, the operations with row 2 will not affect row 1.

Working with one row at a time in the above manner, it i s clear that

in a finite number of steps we will arrive at a row-reduced matrix. I

Exercises

1. Find all solutions to the system of equations

2. If

(1 - i)Xl - iX2 =^0 2Xl + (1 i)X2 O.

find all solutions of AX = 0 by row-reducing (^) A.

Sec. 1.

3. If

A =^ [ � =� �J

• 1 ° 3

Row-Reduced Echelon Matrices

find all solutions of AX = 2X and all solutions of AX = 3X. (The symbol eX denotes the matrix each entry of which is c times the corresponding entry of X.)

4. Find a row-reduced matrix which is row-equivalent to

A =

[l

( 1 + i)

• 2

2i -1�J'

5. Prove that the following two matrices are not row-equivalent:

[� -! �J [-� � -rJ

6. Let

A = [; �J

be a 2 X 2 matrix with complex entries. Suppose that A is row-reduced and also that a + b + c + d 0. Prove that there are exactly three such matrices.

7. Prove that the interchange of two rows of a matrix can be accomplished by a finite sequence of elementary row operations of the other two types.
8. Consider the system of equations AX = ° where

A = [; �J

is a 2 X 2 matrix over the field F. Prove the following. (a) If every entry of A is 0, then every pair (Xl, X2) is a solution of AX = 0. (b) If ad - be ¢ 0, the system AX ° has only the trivial solution Xl X2 =^ 0. (c) If ad - be = ° and some entry of A is different from 0, then there is a solution (x�, xg) such that (Xl, X2) is a solution if and only if there is a scalar y such that Xl = yx?, X2 =^ yxg.

11

1 .4. Row-Reduced Echelon Matrices

Until now, our work with systems of linear equations was motivated

by an attempt to find the solutions of such a system. In Section 1.3 we

established a standardized technique for finding these solutions. We wish now to acquire some information which is slightly more theoretical, and for that purpose it is convenient to go a little beyond row-reduced matrices.

Definition. An m X n rnatrix R is called a row-reduced echelon

lllatrix if: