Linear Algebra

Description

Linear Algebra offers a unified treatment of both matrix-oriented and theoretical approaches to the course, which will be useful for classes with a mix of mathematics, physics, engineering, and computer science students. Major topics include singular value decomposition, the spectral theorem, linear systems of equations, vector spaces, linear maps, matrices, eigenvalues and eigenvectors, linear independence, bases, coordinates, dimension, matrix factorizations, inner products, norms, and determinants. Cover Half-title page Series page Title page Copyright page Dedication Contents Preface To the Student 1 Linear Systems and Vector Spaces 1.1 Linear Systems of Equations Bread, Beer, and Barley Linear Systems and Solutions 1.2 Gaussian Elimination The Augmented Matrix of a Linear System Row Operations Does it Always Work? Pivots and Existence and Uniqueness of Solutions 1.3 Vectors and the Geometry of Linear Systems Vectors and Linear Combinations The Vector Form of a Linear System The Geometry of Linear Combinations The Geometry of Solutions 1.4 Fields General Fields Arithmetic in Fields Linear Systems over a Field 1.5 Vector Spaces General Vector Spaces Examples of Vector Spaces Arithmetic in Vector Spaces 2 Linear Maps and Matrices 2.1 Linear Maps Recognizing Sameness Linear Maps in Geometry Matrices as Linear Maps Eigenvalues and Eigenvectors The Matrix–Vector Form of a Linear System 2.2 More on Linear Maps Isomorphism Properties of Linear Maps The Matrix of a Linear Map Some Linear Maps on Function and Sequence Spaces 2.3 Matrix Multiplication Definition of Matrix Multiplication Other Ways of Looking at Matrix Multiplication The Transpose Matrix Inverses 2.4 Row Operations and the LU Decomposition Row Operations and Matrix Multiplication Inverting Matrices via Row Operations The LU Decomposition 2.5 Range, Kernel, and Eigenspaces Range Kernel Eigenspaces Solution Spaces 2.6 Error-correcting Linear Codes Linear Codes Error-detecting Codes Error-correcting Codes The Hamming Code 3 Linear Independence, Bases, and Coordinates 3.1 Linear (In)dependence Redundancy Linear Independence The Linear Dependence Lemma Linear Independence of Eigenvectors 3.2 Bases Bases of Vector Spaces Properties of Bases Bases and Linear Maps 3.3 Dimension The Dimension of a Vector Space Dimension, Bases, and Subspaces 3.4 Rank and Nullity The Rank and Nullity of Maps and Matrices The Rank–Nullity Theorem Consequences of the Rank–Nullity Theorem Linear Constraints 3.5 Coordinates Coordinate Representations of Vectors Matrix Representations of Linear Maps Eigenvectors and Diagonalizability Matrix Multiplication and Coordinates 3.6 Change of Basis Change of Basis Matrices Similarity and Diagonalizability Invariants 3.7 Triangularization Eigenvalues of Upper Triangular Matrices Triangularization 4 Inner Products 4.1 Inner Products The Dot Product in R[sup(n)] Inner Product Spaces Orthogonality More Examples of Inner Product Spaces 4.2 Orthonormal Bases Orthonormality Coordinates in Orthonormal Bases The Gram–Schmidt Process 4.3 Orthogonal Projections and Optimization Orthogonal Complements and Direct Sums Orthogonal Projections Linear Least Squares Approximation of Functions 4.4 Normed Spaces General Norms The Operator Norm 4.5 Isometries Preserving Lengths and Angles Orthogonal and Unitary Matrices The QR Decomposition 5 Singular Value Decomposition and the Spectral Theorem 5.1 Singular Value Decomposition of Linear Maps Singular Value Decomposition Uniqueness of Singular Values 5.2 Singular Value Decomposition of Matrices Matrix Version of SVD SVD and Geometry Low-rank Approximation 5.3 Adjoint Maps The Adjoint of a Linear Map Self-adjoint Maps and Matrices The Four Subspaces Computing SVD 5.4 The Spectral Theorems Eigenvectors of Self-adjoint Maps and Matrices Normal Maps and Matrices Schur Decomposition 6 Determinants 6.1 Determinants Multilinear Functions The Determinant Existence and Uniqueness of the Determinant 6.2 Computing Determinants Basic Properties Determinants and Row Operations Permutations 6.3 Characteristic Polynomials The Characteristic Polynomial of a Matrix Multiplicities of Eigenvalues The Cayley–Hamilton Theorem 6.4 Applications of Determinants Volume Cramer's Rule Cofactors and Inverses Appendix A.1 Sets and Functions Basic Definitions Composition and Invertibility A.2 Complex Numbers A.3 Proofs Logical Connectives Quantifiers Contrapositives, Counterexamples, and Proof by Contradiction Proof by Induction Addendum Hints and Answers to Selected Exercises Index