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Linear Algebra Done Right, Second Edition - UFPE

Linear AlgebraDone Right, Second EditionSheldon AxlerSpringerContentsPreface to the InstructorixPreface to the StudentxiiiAcknowledgmentsxvChapter1 Vector Spaces1 Complex of Vector of Vector and Direct Vector Spaces21 Span and Linear Maps37 Definitions and Spaces and Matrix of a Linear and Eigenvectors75 Invariant Applied to Subspaces on Real Vector Spaces97 Inner Projections and Minimization Functionals and on Inner-Product Spaces127 Self-Adjoint and Normal Spectral Operators on Real Inner-Product and Singular-Value on Complex Vector Spaces163 Generalized Characteristic of an Minimal on Real Vector Spaces193 Eigenvalues of Square Upper-Triangular Characteristic and Determinant213 Change of of an of a Index247 Index249 Preface to the InstructorYou are probably about to teach a

A goal more important than teaching any particular set of theorems is to develop in students the ability to understand and manipulate the objects of linear algebra. Mathematics can be learned only by doing; fortunately, linear algebra has many good homework problems. When teaching this course, I usually assign two or three of the exercises each

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Transcription of Linear Algebra Done Right, Second Edition - UFPE

1 Linear AlgebraDone Right, Second EditionSheldon AxlerSpringerContentsPreface to the InstructorixPreface to the StudentxiiiAcknowledgmentsxvChapter1 Vector Spaces1 Complex of Vector of Vector and Direct Vector Spaces21 Span and Linear Maps37 Definitions and Spaces and Matrix of a Linear and Eigenvectors75 Invariant Applied to Subspaces on Real Vector Spaces97 Inner Projections and Minimization Functionals and on Inner-Product Spaces127 Self-Adjoint and Normal Spectral Operators on Real Inner-Product and Singular-Value on Complex Vector Spaces163 Generalized Characteristic of an Minimal on Real Vector Spaces193 Eigenvalues of Square Upper-Triangular Characteristic and Determinant213 Change of of an of a Index247 Index249 Preface to the InstructorYou are probably about to teach a

2 Course that will give studentstheir Second exposure to Linear Algebra . During their first brush withthe subject, your students probably worked with Euclidean spaces andmatrices. In contrast, this course will emphasize abstract vector spacesand Linear audacious title of this book deserves an explanation. Almostall Linear Algebra books use determinants to prove that every Linear op-erator on a finite-dimensional complex vector space has an are difficult, nonintuitive, and often defined without mo-tivation. To prove the theorem about existence of eigenvalues on com-plex vector spaces, most books must define determinants, prove that alinear map is not invertible if and only if its determinant equals 0, andthen define the characteristic polynomial. This tortuous (torturous?)path gives students little feeling for why eigenvalues must contrast, the simple determinant-free proofs presented here of-fer more insight.

3 Once determinants have been banished to the endof the book, a new route opens to the main goal of Linear Algebra understanding the structure of Linear book starts at the beginning of the subject, with no prerequi-sites other than the usual demand for suitable mathematical if your students have already seen some of the material in thefirst few chapters, they may be unaccustomed to working exercises ofthe type presented here, most of which require an understanding ofproofs. Vector spaces are defined in Chapter 1, and their basic propertiesare developed. Linear independence, span, basis, and dimension are defined inChapter 2, which presents the basic theory of finite-dimensionalvector to the Instructor Linear maps are introduced in Chapter 3. The key result hereis that for a Linear mapT, the dimension of the null space ofTplus the dimension of the range ofTequals the dimension of thedomain ofT.

4 The part of the theory of polynomials that will be needed to un-derstand Linear operators is presented in Chapter 4. If you takeclass time going through the proofs in this chapter (which con-tains no Linear Algebra ), then you probably will not have time tocover some important aspects of Linear Algebra . Your studentswill already be familiar with the theorems about polynomials inthis chapter, so you can ask them to read the statements of theresults but not the proofs. The curious students will read someof the proofs anyway, which is why they are included in the text. The idea of studying a Linear operator by restricting it to smallsubspaces leads in Chapter 5 to eigenvectors. The highlight of thechapter is a simple proof that on complex vector spaces, eigenval-ues always exist. This result is then used to show that each linearoperator on a complex vector space has an upper-triangular ma-trix with respect to some basis.

5 Similar techniques are used toshow that every Linear operator on a real vector space has an in-variant subspace of dimension 1 or 2. This result is used to provethat every Linear operator on an odd-dimensional real vector spacehas an eigenvalue. All this is done without defining determinantsor characteristic polynomials! Inner-product spaces are defined in Chapter 6, and their basicproperties are developed along with standard tools such as ortho-normal bases, the Gram-Schmidt procedure, and adjoints. Thischapter also shows how orthogonal projections can be used tosolve certain minimization problems. The spectral theorem, which characterizes the Linear operators forwhich there exists an orthonormal basis consisting of eigenvec-tors, is the highlight of Chapter 7. The work in earlier chapterspays off here with especially simple proofs.

6 This chapter alsodeals with positive operators, Linear isometries, the polar decom-position, and the singular-value to the Instructorxi 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 ona complex vector space in terms of its generalized description enables one to prove almost all the results usu-ally proved using Jordan form. For example, these tools are usedto prove that every invertible Linear operator on a complex vectorspace has a square root. The chapter concludes with a proof thatevery Linear operator on a complex vector space can be put intoJordan form. Linear operators on real vector spaces occupy center stage inChapter 9. Here two-dimensional invariant subspaces make upfor the possible lack of eigenvalues, leading to results analogousto those obtained on complex vector spaces.

7 The trace and determinant are defined in Chapter 10 in termsof 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 possiblewith the traditional approach to eigenvalues because that methoduses determinants to prove that eigenvalues exist. The standardtheorems 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 thedeterminant there seem book usually develops Linear Algebra simultaneously for realand complex vector spaces by lettingFdenote either the real or thecomplex numbers.

8 Abstract fields could be used instead, but to do sowould introduce extra abstraction without leading to any new Linear al-gebra. Another reason for restricting attention to the real and complexnumbers is that polynomials can then be thought of as genuine func-tions instead of the more formal objects needed for polynomials withcoefficients in finite fields. Finally, even if the beginning part of the the-ory were developed with arbitrary fields, inner-product spaces wouldpush consideration back to just real and complex vector to the InstructorEven in a book as short as this one, you cannot expect to cover every-thing. Going through the first eight chapters is an ambitious goal for aone-semester course. If you must reach Chapter 10, then I suggest cov-ering Chapters 1, 2, and 4 quickly (students may have seen this materialin earlier courses) and skipping Chapter 9 (in which case you shoulddiscuss trace and determinants only on complex vector spaces).

9 A goal more important than teaching any particular set of theoremsis to develop in students the ability to understand and manipulate theobjects of Linear Algebra . Mathematics can be learned only by doing;fortunately, Linear Algebra has many good homework problems. Whenteaching this course, I usually assign two or three of the exercises eachclass, due the next class. Going over the homework might take up athird or even half of a typical solutions manual for all the exercises is available (without charge)only to instructors who are using this book as a textbook. To obtainthe solutions manual, instructors should send an e-mail request to me(or contact Springer if I am no longer around).Please check my web site for a list of errata (which I hope will beempty or almost empty) and other information about this would greatly appreciate hearing about any errors in this book,even minor ones.

10 I welcome your suggestions for improvements, eventiny ones. Please feel free to contact fun!Sheldon AxlerMathematics DepartmentSan Francisco State UniversitySan Francisco, CA 94132, home page: to the StudentYou 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 vectorspaces and Linear maps. These terms will be defined later, so don tworry if you don t know what they mean. This book starts from the be-ginning of the subject, assuming no knowledge of Linear Algebra . Thekey point is that you are about to immerse yourself in serious math-ematics, with an emphasis on your attaining a deep understanding ofthe definitions, theorems, and cannot expect to read mathematics the way you read a novel.


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