Exercises and Problems in Linear Algebra

This book contains an extensive collection of exercises and problems that address relevant topics in linear algebra. Topics that the author finds missing or inadequately covered in most existing books are also included. The exercises will be both interesting and helpful to an average student. Some are fairly routine calculations, while others require serious thought.The format of the questions makes them suitable for teachers to use in quizzes and assigned homework. Some of the problems may provide excellent topics for presentation and discussions. Furthermore, answers are given for all odd-numbered exercises which will be extremely useful for self-directed learners. In each chapter, there is a short background section which includes important definitions and statements of theorems to provide context for the following exercises and problems.Contents: PrefaceMatrices and Linear Equations:Arithmetic of MatricesElementary Matrices; DeterminantsVector Geometry in ℝⁿVector Spaces:Vector SpacesSubspacesLinear IndependenceBasis for a Vector SpaceLinear Maps Between Vector Spaces:LinearityLinear Maps Between Euclidean SpacesProjection OperatorsSpectral Theory of Vector Spaces:Eigenvalues and EigenvectorsDiagonalization of MatricesSpectral Theorem for Vector SpacesSome Applications of the Spectral TheoremEvery Operator is Diagonalizable Plus NilpotentThe Geometry of Inner Product Spaces:Complex ArithmeticReal and Complex Inner Product SpacesOrthonormal Sets of VectorsQuadratic FormsOptimizationAdjoint Operators:Adjoints and TransposesThe Four Fundamental SubspacesOrthogonal ProjectionsLeast Squares ApproximationSpectral Theory of Inner Product Spaces:Spectral Theorem for Real Inner Product SpacesSpectral Theorem for Complex Inner Product SpacesBibliographyIndex
Readership: Students and teachers of linear algebra.Linear Algebra;Algebra;Vectors;Vector Spaces;Subspaces; Dependence;Span;Basis;Linear;Transformation;Kernel;Null Space;Eigenvalues;Eigenvectors;Spectrum;Spectral Mapping;Spectral Theorem;Matrices;Diagonalizable;Nilpotent;Inner Product;Gram-schmidt;Orthogonal;Perpendicular;Cauchy-schwarz;Norm;Unitary;Adjoint;Self-adjoint;Hermitian;Diagonalization;Dimension;Real;Complex;Nilpotent;Jordan;Normal Form;Normal;Symmetric0Key Features:Exercises and problems addressing topics missing or inadequately covered in most existing textsEasy-to-grade exercises. Solutions to odd-numbered exercisesInclusion of the LaTeX code on the WC website for all problems and exercisesIncludes important definitions and statements of theorems as backgroundMany of the problems are suitable for group projects or as a basis for lectures