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• Chapter Acknowledgments xv
• Vector Spaces
  • Complex Numbers
  • Definition of Vector Space
  • Properties of Vector Spaces
  • Subspaces
  • Sums and Direct Sums
  • Exercises
• Chapter
• Finite-Dimensional Vector Spaces
  • Span and Linear Independence
  • Bases
  • Dimension
  • Exercises
• Chapter
• Linear Maps
  • Definitions and Examples
  • Null Spaces and Ranges
  • The Matrix of a Linear Map
  • Invertibility
  • Exercises
• Chapter vi Contents
• Polynomials
  • Degree
  • Complex Coefficients
  • Real Coefficients
  • Exercises
• Chapter
• Eigenvalues and Eigenvectors
  • Invariant Subspaces
  • Polynomials Applied to Operators
  • Upper-Triangular Matrices
  • Diagonal Matrices
  • Invariant Subspaces on Real Vector Spaces
  • Exercises
• Chapter
• Inner-Product Spaces
  • Inner Products
  • Norms
  • Orthonormal Bases
  • Orthogonal Projections and Minimization Problems
  • Linear Functionals and Adjoints
  • Exercises
• Chapter
• Operators on Inner-Product Spaces
  • Self-Adjoint and Normal Operators
  • The Spectral Theorem
  • Normal Operators on Real Inner-Product Spaces
  • Positive Operators
  • Isometries
  • Polar and Singular-Value Decompositions
  • Exercises
• Chapter
• Operators on Complex Vector Spaces
  • Generalized Eigenvectors
  • The Characteristic Polynomial
  • Decomposition of an Operator
  • Square Roots Contents vii
  • The Minimal Polynomial
  • Jordan Form
  • Exercises
• Chapter
• Operators on Real Vector Spaces
  • Eigenvalues of Square Matrices
  • Block Upper-Triangular Matrices
  • The Characteristic Polynomial
  • Exercises
• Chapter
• Trace and Determinant
  • Change of Basis
  • Trace
  • Determinant of an Operator
  • Determinant of a Matrix
  • Volume
  • Exercises
• Symbol Index
• Index

x Preface to the Instructor

• Linear maps are introduced in Chapter 3. The key result here is that for a linear map T , the dimension of the null space of T plus the dimension of the range of T equals the dimension of the domain of T.
• The part of the theory of polynomials that will be needed to un- derstand linear operators is presented in Chapter 4. If you take class time going through the proofs in this chapter (which con- tains no linear algebra), then you probably will not have time to cover some important aspects of linear algebra. Your students will already be familiar with the theorems about polynomials in this chapter, so you can ask them to read the statements of the results but not the proofs. The curious students will read some of the proofs anyway, which is why they are included in the text.
• The idea of studying a linear operator by restricting it to small subspaces leads in Chapter 5 to eigenvectors. The highlight of the chapter is a simple proof that on complex vector spaces, eigenval- ues always exist. This result is then used to show that each linear operator on a complex vector space has an upper-triangular ma- trix with respect to some basis. Similar techniques are used to show that every linear operator on a real vector space has an in- variant subspace of dimension 1 or 2. This result is used to prove that every linear operator on an odd-dimensional real vector space has an eigenvalue. All this is done without defining determinants or characteristic polynomials!
• Inner-product spaces are defined in Chapter 6, and their basic properties are developed along with standard tools such as ortho- normal bases, the Gram-Schmidt procedure, and adjoints. This chapter also shows how orthogonal projections can be used to solve certain minimization problems.
• The spectral theorem, which characterizes the linear operators for which there exists an orthonormal basis consisting of eigenvec- tors, is the highlight of Chapter 7. The work in earlier chapters pays off here with especially simple proofs. This chapter also deals with positive operators, linear isometries, the polar decom- position, and the singular-value decomposition.

Preface to the Instructor xi

• The minimal polynomial, characteristic polynomial, and general- ized eigenvectors are introduced in Chapter 8. The main achieve- ment of this chapter is the description of a linear operator on a complex vector space in terms of its generalized eigenvectors. This description enables one to prove almost all the results usu- ally proved using Jordan form. For example, these tools are used to prove that every invertible linear operator on a complex vector space has a square root. The chapter concludes with a proof that every linear operator on a complex vector space can be put into Jordan form.
• Linear operators on real vector spaces occupy center stage in Chapter 9. Here two-dimensional invariant subspaces make up for the possible lack of eigenvalues, leading to results analogous to those obtained on complex vector spaces.
• The trace and determinant are defined in Chapter 10 in terms of the characteristic polynomial (defined earlier without determi- nants). On complex vector spaces, these definitions can be re- stated: the trace is the sum of the eigenvalues and the determi- nant is the product of the eigenvalues (both counting multiplic- ity). These easy-to-remember definitions would not be possible with the traditional approach to eigenvalues because that method uses determinants to prove that eigenvalues exist. The standard theorems about determinants now become much clearer. The po- lar decomposition and the characterization of self-adjoint opera- tors are used to derive the change of variables formula for multi- variable integrals in a fashion that makes the appearance of the determinant there seem natural.

This book usually develops linear algebra simultaneously for real and complex vector spaces by letting F denote either the real or the complex numbers. Abstract fields could be used instead, but to do so would introduce extra abstraction without leading to any new linear al- gebra. Another reason for restricting attention to the real and complex numbers is that polynomials can then be thought of as genuine func- tions instead of the more formal objects needed for polynomials with coefficients in finite fields. Finally, even if the beginning part of the the- ory were developed with arbitrary fields, inner-product spaces would push consideration back to just real and complex vector spaces.

Preface to the Student

You are probably about to begin your second exposure to linear al- gebra. Unlike your first brush with the subject, which probably empha- sized Euclidean spaces and matrices, we will focus on abstract vector spaces and linear maps. These terms will be defined later, so don’t worry if you don’t know what they mean. This book starts from the be- ginning of the subject, assuming no knowledge of linear algebra. The key point is that you are about to immerse yourself in serious math- ematics, with an emphasis on your attaining a deep understanding of the definitions, theorems, and proofs. You cannot expect to read mathematics the way you read a novel. If you zip through a page in less than an hour, you are probably going too fast. When you encounter the phrase “as you should verify”, you should indeed do the verification, which will usually require some writing on your part. When steps are left out, you need to supply the missing pieces. You should ponder and internalize each definition. For each theorem, you should seek examples to show why each hypothesis is necessary. Please check my web site for a list of errata (which I hope will be empty or almost empty) and other information about this book. I would greatly appreciate hearing about any errors in this book, even minor ones. I welcome your suggestions for improvements, even tiny ones. Have fun!

Sheldon Axler Mathematics Department San Francisco State University San Francisco, CA 94132, USA

e-mail: [email protected] www home page: http://math.sfsu.edu/axler

xiii

Acknowledgments

I owe a huge intellectual debt to the many mathematicians who cre- ated linear algebra during the last two centuries. In writing this book I tried to think about the best way to present linear algebra and to prove its theorems, without regard to the standard methods and proofs used in most textbooks. Thus I did not consult other books while writing this one, though the memory of many books I had studied in the past surely influenced me. Most of the results in this book belong to the common heritage of mathematics. A special case of a theorem may first have been proved in antiquity (which for linear algebra means the nineteenth century), then slowly sharpened and improved over decades by many mathematicians. Bestowing proper credit on all the contrib- utors would be a difficult task that I have not undertaken. In no case should the reader assume that any theorem presented here represents my original contribution. Many people helped make this a better book. For useful sugges- tions and corrections, I am grateful to William Arveson (for suggesting the proof of 5.13), Marilyn Brouwer, William Brown, Robert Burckel, Paul Cohn, James Dudziak, David Feldman (for suggesting the proof of 8.40), Pamela Gorkin, Aram Harrow, Pan Fong Ho, Dan Kalman, Robert Kantrowitz, Ramana Kappagantu, Mizan Khan, Mikael Lindstr¨om, Ja- cob Plotkin, Elena Poletaeva, Mihaela Poplicher, Richard Potter, Wade Ramey, Marian Robbins, Jonathan Rosenberg, Joan Stamm, Thomas Starbird, Jay Valanju, and Thomas von Foerster. Finally, I thank Springer for providing me with help when I needed it and for allowing me the freedom to make the final decisions about the content and appearance of this book.

xv

2 Chapter 1. Vector Spaces

Complex Numbers

You should already be familiar with the basic properties of the set R of real numbers. Complex numbers were invented so that we can take square roots of negative numbers. The key idea is to assume we have The symbol i was first a square root of −1, denoted i , and manipulate it using the usual rules used to denote

− 1 by the Swiss mathematician Leonhard Euler in 1777.

of arithmetic. Formally, a complex number is an ordered pair (a, b) , where a, b ∈ R, but we will write this as a + bi. The set of all complex numbers is denoted by C:

C = { a + bi : a, b ∈ R}.

If a ∈ R, we identify a + 0 i with the real number a. Thus we can think of R as a subset of C. Addition and multiplication on C are defined by

(a + bi) + (c + di) = (a + c) + (b + d)i, (a + bi)(c + di) = (ac − bd) + (ad + bc)i ;

here a, b, c, d ∈ R. Using multiplication as defined above, you should verify that i^2 = −1. Do not memorize the formula for the product of two complex numbers; you can always rederive it by recalling that i^2 = −1 and then using the usual rules of arithmetic. You should verify, using the familiar properties of the real num- bers, that addition and multiplication on C satisfy the following prop- erties: commutativity w + z = z + w and wz = zw for all w, z ∈ C;

associativity (z 1 + z 2 ) + z 3 = z 1 + (z 2 + z 3 ) and (z 1 z 2 )z 3 = z 1 (z 2 z 3 ) for all z 1 , z 2 , z 3 ∈ C;

identities z + 0 = z and z 1 = z for all z ∈ C;

additive inverse for every z ∈ C, there exists a unique w ∈ C such that z + w = 0;

multiplicative inverse for every z ∈ C with z = 0, there exists a unique w ∈ C such that zw = 1;

Complex Numbers 3

distributive property λ(w + z) = λw + λz for all λ, w, z ∈ C.

For z ∈ C, we let − z denote the additive inverse of z. Thus − z is the unique complex number such that

z + ( − z) = 0_._

Subtraction on C is defined by

w − z = w + ( − z)

for w, z ∈ C. For z ∈ C with z = 0, we let 1 /z denote the multiplicative inverse of z. Thus 1 /z is the unique complex number such that

z( 1 /z) = 1_._

Division on C is defined by

w/z = w( 1 /z)

for w, z ∈ C with z = 0. So that we can conveniently make definitions and prove theorems that apply to both real and complex numbers, we adopt the following notation:

The letter F is used because R and C are examples of what are called fields. In this book we will not need to deal with fields other than R or C_. Many of the definitions, theorems, and proofs in linear algebra that work for both_ R and C also work without change if an arbitrary field replaces R or C_._

Throughout this book, F stands for either R or C.

Thus if we prove a theorem involving F, we will know that it holds when F is replaced with R and when F is replaced with C. Elements of F are called scalars. The word “scalar”, which means number, is often used when we want to emphasize that an object is a number, as opposed to a vector (vectors will be defined soon). For z ∈ F and m a positive integer, we define z m^ to denote the product of z with itself m times:

z m^ = z ︸ · · · · ·︷︷ z ︸ m times

Clearly (z m^ )n^ = z mn^ and (wz)m^ = w m^ z m^ for all w, z ∈ F and all positive integers m, n.

Definition of Vector Space 5

do not have the same length), though the sets { 4 , 4 } and { 4 , 4 , 4 } both equal the set { 4 }. To define the higher-dimensional analogues of R^2 and R^3 , we will simply replace R with F (which equals R or C) and replace the 2 or 3 with an arbitrary positive integer. Specifically, fix a positive integer n for the rest of this section. We define F n^ to be the set of all lists of length n consisting of elements of F:

F n^ = { (x 1 ,... , xn ) : xj ∈ F for j = 1 ,... , n }.

For example, if F = R and n equals 2 or 3, then this definition of F n agrees with our previous notions of R^2 and R^3. As another example, C^4 is the set of all lists of four complex numbers:

C^4 = { (z 1 , z 2 , z 3 , z 4 ) : z 1 , z 2 , z 3 , z 4 ∈ C}.

If n ≥ 4, we cannot easily visualize R n^ as a physical object. The same For an amusing account of how R^3 would be perceived by a creature living in R^2 , read Flatland: A Romance of Many Dimensions, by Edwin A. Abbott. This novel, published in 1884, can help creatures living in three-dimensional space, such as ourselves, imagine a physical space of four or more dimensions.

problem arises if we work with complex numbers: C^1 can be thought of as a plane, but for n ≥ 2, the human brain cannot provide geometric models of C n. However, even if n is large, we can perform algebraic manipulations in F n^ as easily as in R^2 or R^3. For example, addition is defined on F n^ by adding corresponding coordinates:

1.1 (x 1 ,... , xn ) + (y 1 ,... , yn ) = (x 1 + y 1 ,... , xn + yn ).

Often the mathematics of F n^ becomes cleaner if we use a single entity to denote an list of n numbers, without explicitly writing the coordinates. Thus the commutative property of addition on F n^ should be expressed as x + y = y + x

for all x, y ∈ F n , rather than the more cumbersome

(x 1 ,... , xn ) + (y 1 ,... , yn ) = (y 1 ,... , yn ) + (x 1 ,... , xn )

for all x 1 ,... , xn , y 1 ,... , yn ∈ F (even though the latter formulation is needed to prove commutativity). If a single letter is used to denote an element of F n , then the same letter, with appropriate subscripts, is often used when coordinates must be displayed. For example, if x ∈ F n , then letting x equal (x 1 ,... , xn ) is good notation. Even better, work with just x and avoid explicit coordinates, if possible.

6 Chapter 1. Vector Spaces

We let 0 denote the list of length n all of whose coordinates are 0:

0 = ( 0 ,... , 0 ).

Note that we are using the symbol 0 in two different ways—on the left side of the equation above, 0 denotes a list of length n , whereas on the right side, each 0 denotes a number. This potentially confusing practice actually causes no problems because the context always makes clear what is intended. For example, consider the statement that 0 is an additive identity for F n :

x + 0 = x

for all x ∈ F n. Here 0 must be a list because we have not defined the sum of an element of F n^ (namely, x ) and the number 0. A picture can often aid our intuition. We will draw pictures de- picting R^2 because we can easily sketch this space on two-dimensional surfaces such as paper and blackboards. A typical element of R^2 is a point x = (x 1 , x 2 ). Sometimes we think of x not as a point but as an arrow starting at the origin and ending at (x 1 , x 2 ) , as in the picture below. When we think of x as an arrow, we refer to it as a vector.

x (^) 1 -axis

x (^) 2 -axis

( x 1 , x 2 )

x

Elements of R^2 can be thought of as points or as vectors.

The coordinate axes and the explicit coordinates unnecessarily clut- ter the picture above, and often you will gain better understanding by dispensing with them and just thinking of the vector, as in the next picture.

8 Chapter 1. Vector Spaces

Having dealt with addition in F n , we now turn to multiplication. We could define a multiplication on F n^ in a similar fashion, starting with two elements of F n^ and getting another element of F n^ by multiplying corresponding coordinates. Experience shows that this definition is not useful for our purposes. Another type of multiplication, called scalar multiplication, will be central to our subject. Specifically, we need to define what it means to multiply an element of F n^ by an element of F. We make the obvious definition, performing the multiplication in each coordinate: a(x 1 ,... , xn ) = (ax 1 ,... , ax (^) n ) ; here a ∈ F and (x 1 ,... , xn ) ∈ F n. In scalar multiplication, Scalar multiplication has a nice geometric interpretation in R^2. If we multiply together a scalar and a vector, getting a vector. You may be familiar with the dot product in R^2 or R^3 , in which we multiply together two vectors and obtain a scalar. Generalizations of the dot product will become important when we study inner products in Chapter 6. You may also be familiar with the cross product in R^3 , in which we multiply together two vectors and obtain another vector. No useful generalization of this type of multiplication exists in higher dimensions.

a is a positive number and x is a vector in R^2 , then ax is the vector that points in the same direction as x and whose length is a times the length of x. In other words, to get ax , we shrink or stretch x by a factor of a , depending upon whether a < 1 or a > 1. The next picture illustrates this point.

x (1/2) x

(3/2) x

Multiplication by positive scalars

If a is a negative number and x is a vector in R^2 , then ax is the vector that points in the opposite direction as x and whose length is | a | times the length of x , as illustrated in the next picture.

x

(−1/2) x

(−3/2) x

Multiplication by negative scalars

Definition of Vector Space 9

The motivation for the definition of a vector space comes from the important properties possessed by addition and scalar multiplication on F n. Specifically, addition on F n^ is commutative and associative and has an identity, namely, 0. Every element has an additive inverse. Scalar multiplication on F n^ is associative, and scalar multiplication by 1 acts as a multiplicative identity should. Finally, addition and scalar multi- plication on F n^ are connected by distributive properties. We will define a vector space to be a set V along with an addition and a scalar multiplication on V that satisfy the properties discussed in the previous paragraph. By an addition on V we mean a function that assigns an element u + v ∈ V to each pair of elements u, v ∈ V. By a scalar multiplication on V we mean a function that assigns an element av ∈ V to each a ∈ F and each v ∈ V. Now we are ready to give the formal definition of a vector space. A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold:

commutativity u + v = v + u for all u, v ∈ V ;

associativity (u + v) + w = u + (v + w) and (ab)v = a(bv) for all u, v, w ∈ V and all a, b ∈ F;

additive identity there exists an element 0 ∈ V such that v + 0 = v for all v ∈ V ;

additive inverse for every v ∈ V , there exists w ∈ V such that v + w = 0;

multiplicative identity 1 v = v for all v ∈ V ;

distributive properties a(u + v) = au + av and (a + b)u = au + bu for all a, b ∈ F and all u, v ∈ V.

The scalar multiplication in a vector space depends upon F. Thus when we need to be precise, we will say that V is a vector space over F instead of saying simply that V is a vector space. For example, R n^ is a vector space over R, and C n^ is a vector space over C. Frequently, a vector space over R is called a real vector space and a vector space over