Linear Algebra

About our authors Stephen H. Friedberg holds a BA in mathematics from Boston University and MS and PhD degrees in mathematics from Northwestern University, and was awarded a Moore Postdoctoral Instructorship at MIT. He served as a director for CUPM, the Mathematical Association of America's Committee on the Undergraduate Program in Mathematics. He was a faculty member at Illinois State University for 32 years, where he was recognized as the outstanding teacher in the College of Arts and Sciences in 1990. He has also taught at the University of London, the University of Missouri and at Illinois Wesleyan University. He has authored or coauthored articles and books in analysis and linear algebra.Arnold J. Insel received BA and MA degrees in mathematics from the University of Florida and a PhD from the University of California at Berkeley. He served as a faculty member at Illinois State University for 31 years and at Illinois Wesleyan University for 2 years. In addition to authoring and co-authoring articles and books in linear algebra, he has written articles in lattice theory, topology and topological groups.Lawrence E. Spence holds a BA from Towson State College and MS and PhD degrees in mathematics from Michigan State University. He served as a faculty member at Illinois State University for 34 years, where he was recognized as the outstanding teacher in the College of Arts and Sciences in 1987. He is an author or co-author of 9 college mathematics textbooks, as well as articles in mathematics journals in the areas of discrete mathematics and linear algebra.

• * Sections denoted by an asterisk are optional. Vector Spaces 1.1 Introduction1.2 Vector Spaces1.3 Subspaces1.4 Linear Combinations and Systems of Linear Equations1.5 Linear Dependence and Linear Independence1.6 Bases and Dimension1.7* Maximal Linearly Independent Subsets Index of DefinitionsLinear Transformations and Matrices 2.1 Linear Transformations, Null Spaces, and Ranges2.2 The Matrix Representation of a Linear Transformation2.3 Composition of Linear Transformations and Matrix Multiplication2.4 Invertibility and Isomorphisms2.5 The Change of Coordinate Matrix2.6* Dual Spaces2.7* Homogeneous Linear Differential Equations with Constant Coefficients Index of DefinitionsElementary Matrix Operations and Systems of Linear Equations 3.1 Elementary Matrix Operations and Elementary Matrices3.2 The Rank of a Matrix and Matrix Inverses3.3 Systems of Linear Equations – Theoretical Aspects3.4 Systems of Linear Equations – Computational Aspects Index of DefinitionsDeterminants 4.1 Determinants of Order 24.2 Determinants of Order n4.3 Properties of Determinants4.4 Summary|Important Facts about Determinants4.5* A Characterization of the Determinant Index of DefinitionsDiagonalization 5.1 Eigenvalues and Eigenvectors5.2 Diagonalizability5.3* Matrix Limits and Markov Chains5.4 Invariant Subspaces and the Cayley–Hamilton Theorem Index of DefinitionsInner Product Spaces 6.1 Inner Products and Norms6.2 The Gram–Schmidt Orthogonalization Process and Orthogonal Complements6.3 The Adjoint of a Linear Operator6.4 Normal and Self-Adjoint Operators6.5 Unitary and Orthogonal Operators and Their Matrices6.6 Orthogonal Projections and the Spectral Theorem6.7* The Singular Value Decomposition and the Pseudoinverse6.8* Bilinear and Quadratic Forms6.9* Einstein's Special Theory of Relativity6.10* Conditioning and the Rayleigh Quotient6.11* The Geometry of Orthogonal Operators Index of DefinitionsCanonical Forms 7.1 The Jordan Canonical Form I7.2 The Jordan Canonical Form II7.3 The Minimal Polynomial7.4* The Rational Canonical Form Index of DefinitionsAppendices A. SetsB. FunctionsC. FieldsD. Complex NumbersE. PolynomialsAnswers to Selected ExercisesIndex