Linear Algebra Abridged

1 Linear Algebra AbridgedSheldon AxlerThis file is generated fromLinear Algebra Done Right(third edition) byexcluding all proofs, examples, and exercises, along with most Linear Algebra without proofs, examples, and exercises is probablyimpossible. Thus this Abridged version should not substitute for the full , this Abridged version may be useful to students seeking to reviewthe statements of the main results of Linear a visual aid, definitions are in beige boxes and theorems are in blueboxes. The numbering of definitions and theorems is the same as in the fullbook. Thus is followed in this Abridged version by (the missing to an example in the full version that is not present here).

2 This file is available without charge. Users have permission to read this filefreely on electronic devices but do not have permission to print full version ofLinear Algebra Done Rightis available at in both printed and electronic forms. A free sample chapterof the full version, and other information, is available at the book s website: March 2016 2015 Contents1 Vector Numbers2 Lists4Fn4 Digression on Definition of Vector Subspaces11 Sums of Subspaces12 Direct Sums132 Finite-Dimensional Vector Span and Linear Independence15 Linear Combinations and Span15 Linear Dimension213 Linear The Vector Space of Linear Maps24 Definition and Examples of Linear Maps24 Algebraic Operations ;W /24 Linear Algebra Abridgedis generated fromLinear Algebra Done Right(by Sheldon Axler, third edition)by excluding all proofs, examples, and exercises, along with most comments.

3 The full version ofLinearAlgebra Done Rightis available at and in both printed and electronic Null Spaces and Ranges26 Null Space and Injectivity26 Range and Surjectivity27 Fundamental Theorem of Linear Matrices29 Representing a Linear Map by a Matrix29 Addition and Scalar Multiplication of Matrices30 Matrix Invertibility and Isomorphic Vector Spaces35 Invertible Linear Maps35 Isomorphic Vector Spaces36 Linear Maps Thought of as Matrix Products and Quotients of Vector Spaces39 Products of Vector Spaces39 Products and Direct Sums40 Quotients of Vector Duality44 The Dual Space and the Dual Map44 The Null Space and Range of the Dual of a Linear Map45 The Matrix of the Dual of a Linear Map47 The Rank of a Matrix484 Polynomials49 Complex Conjugate and Absolute Value50 Uniqueness of Coefficients for Polynomials51 The Division Algorithm for Polynomials52 Zeros of Polynomials52 Factorization of Polynomials overC53 Factorization of Polynomials overR55 Linear Algebra Abridgedis generated fromLinear Algebra Done Right(by Sheldon Axler, third edition)by excluding all proofs, examples, and exercises, along with most comments.

4 The full version ofLinearAlgebra Done Rightis available at and in both printed and electronic , Eigenvectors, and Invariant Invariant Subspaces58 Eigenvalues and Eigenvectors59 Restriction and Quotient Eigenvectors and Upper-Triangular Matrices61 Polynomials Applied to Operators61 Existence of Eigenvalues62 Upper-Triangular Eigenspaces and Diagonal Matrices666 Inner Product Inner Products and Norms70 Inner Orthonormal Bases76 Linear Functionals on Inner Product Orthogonal Complements and Minimization Problems80 Orthogonal Complements80 Minimization Problems827 Operators on Inner Product Self-Adjoint and Normal Operators85 Adjoints85 Self-Adjoint Operators86 Normal The Spectral Theorem89 The Complex Spectral Theorem89 The Real Spectral

5 Positive Operators and Isometries92 Positive Operators92 Isometries93 Linear Algebra Abridgedis generated fromLinear Algebra Done Right(by Sheldon Axler, third edition)by excluding all proofs, examples, and exercises, along with most comments. The full version ofLinearAlgebra Done Rightis available at and in both printed and electronic Decomposition and Singular Value Decomposition94 Polar Decomposition94 Singular Value Decomposition958 Operators on Complex Vector Generalized Eigenvectors and Nilpotent Operators98 Null Spaces of Powers of an Operator98 Generalized Eigenvectors99 Nilpotent Decomposition of an Operator101 Description of Operators on Complex Vector Spaces101 Multiplicity of an Eigenvalue102 Block Diagonal Matrices103 Square Characteristic and Minimal Polynomials105 The Cayley Hamilton Theorem105 The Minimal Jordan Form1079 Operators on Real Vector

6 Complexification110 Complexification of a Vector Space110 Complexification of an Operator111 The Minimal Polynomial of the Complexification112 Eigenvalues of the Complexification112 Characteristic Polynomial of the Operators on Real Inner Product Spaces115 Normal Operators on Real Inner Product Spaces115 Isometries on Real Inner Product Spaces11710 Trace and Trace119 Change of Basis119 Linear Algebra Abridgedis generated fromLinear Algebra Done Right(by Sheldon Axler, third edition)by excluding all proofs, examples, and exercises, along with most comments. The full version ofLinearAlgebra Done Rightis available at and in both printed and electronic : A Connection Between Operators and Determinant123 Determinant of an Operator123 Determinant of a Matrix125 The Sign of the Determinant129 Volume130 Photo Credits135 Index136 Linear Algebra Abridgedis generated fromLinear Algebra Done Right(by Sheldon Axler, third edition)by excluding all proofs, examples, and exercises, along with most comments.

7 The full version ofLinearAlgebra Done Rightis available at and in both printed and electronic Descartes explaining hiswork to Queen Christina ofSweden. Vector spaces are ageneralization of thedescription of a plane usingtwo coordinates, as publishedby Descartes in SpacesLinear Algebra is the study of Linear maps on finite-dimensional vector we will learn what all these terms mean. In this chapter we willdefine vector spaces and discuss their elementary Linear Algebra , better theorems and more insight emerge if complexnumbers are investigated along with real numbers. Thus we will begin byintroducing the complex numbers and their basic will generalize the examples of a plane and ordinary space toRnandCn, which we then will generalize to the notion of a vector space.

8 Theelementary properties of a vector space will already seem familiar to our next topic will be subspaces, which play a role for vector spacesanalogous to the role played by subsets for sets. Finally, we will look at sumsof subspaces (analogous to unions of subsets) and direct sums of subspaces(analogous to unions of disjoint sets).LEARNING OBJECTIVES FOR THIS CHAPTER basic properties of the complex numbersRnandCnvector spacessubspacessums and direct sums of subspacesLinear Algebra Abridgedis generated fromLinear Algebra Done Right(by Sheldon Axler, third edition)by excluding all proofs, examples, and exercises, along with most comments. The full version ofLinearAlgebra Done Rightis available at and in both printed and electronic numbers Acomplex numberis an ordered ;b/, wherea;b2R, butwe will write this asaCbi.

9 The set of all complex numbers is denoted byC:CDfaCbiWa;b2Rg: Addition and multiplicationonCare defined ;.aCbi/. ;b;c; , we identifyaC0iwith the real numbera. Thus we can thinkofRas a subset ofC. We also usually write0 Cbias justbi, and we usuallywrite0C1ias multiplication as defined above, you should verify thati2D not memorize the formula for the product of two complex numbers; youcan always rederive it by recalling thati2D 1and then using the usual rulesof arithmetic (as given by ). of complex arithmeticcommutativity C D C and D for all ; 2C;associativity. C /C D C. C /and. / D . /for all ; ; 2C;identities C0D and 1D for all 2C;additive inversefor every 2C, there exists a unique 2 Csuch that C D0;multiplicative inversefor every 2 Cwith 0, there exists a unique 2 Csuch that D1;distributive property.

10 C /D C for all ; ; Algebra Abridgedis generated fromLinear Algebra Done Right(by Sheldon Axler, third edition)by excluding all proofs, examples, and exercises, along with most comments. The full version ofLinearAlgebra Done Rightis available at and in both printed and electronic properties above are proved using the familiar properties of realnumbers and the definitions of complex addition and , subtraction,1= , divisionLet ; 2C. Let denote the additive inverse of . Thus is the uniquecomplex number such that C. /D0: SubtractiononCis defined by D C. /: For 0, let1= denote the multiplicative inverse of . Thus1= is the unique complex number such that.