Useful Concepts and Results at the Heart of Linear Algebra
A one- or two-semester course for a wide variety of students at the sophomore/junior undergraduate level
A Modern Introduction to Linear Algebra provides a rigorous yet accessible matrix-oriented introduction to the essential concepts of linear algebra. Concrete, easy-to-understand examples motivate the theory.
The book first discusses vectors, Gaussian elimination, and reduced row echelon forms. It then offers a thorough introduction to matrix algebra, including defining the determinant naturally from the PA=LU factorization of a matrix. The author goes on to cover finite-dimensional real vector spaces, infinite-dimensional spaces, linear transformations, and complex vector spaces. The final chapter presents Hermitian and normal matrices as well as quadratic forms.
Taking a computational, algebraic, and geometric approach to the subject, this book provides the foundation for later courses in higher mathematics. It also shows how linear algebra can be used in various areas of application. Although written in a "pencil and paper" manner, the text offers ample opportunities to enhance learning with calculators or computer usage.
Solutions manual available for qualifying instructors
Henry Ricardo is a professor of mathematics at Medgar Evers College of the City University of New York, where he was presented with the 2008 Distinguished Service Award by the School of Science, Health and Technology. Dr. Ricardo was also given the 2009 Distinguished Service Award by the Metropolitan New York Section of the MAA, of which he is the Governor.
Vectors Vectors in Rn The Inner Product and Norm Spanning Sets Linear Independence Bases Subspaces Summary Systems of Equations The Geometry of Systems of Equations in R2 and R3 Matrices and Echelon Form Gaussian Elimination Computational Considerations-Pivoting Gauss-Jordan Elimination and Reduced Row Echelon Form Ill-Conditioned Systems of Linear Equations Rank and Nullity of a Matrix Systems of m Linear Equations in n Unknowns Matrix Algebra Addition and Subtraction of Matrices Matrix-Vector Multiplication The Product of Two Matrices Partitioned Matrices Inverses of Matrices Elementary Matrices The LU Factorization Eigenvalues, Eigenvectors, and Diagonalization Determinants Determinants and Geometry The Manual Calculation of Determinants Eigenvalues and Eigenvectors Similar Matrices and Diagonalization Algebraic and Geometric Multiplicities of Eigenvalues The Diagonalization of Real Symmetric Matrices The Cayley-Hamilton Theorem (a First Look)/the Minimal Polynomial Vector Spaces Vector Spaces Subspaces Linear Independence and the Span Bases and Dimension Linear Transformations Linear Transformations The Range and Null Space of a Linear Transformation The Algebra of Linear Transformations Matrix Representation of a Linear Transformation Invertible Linear Transformations Isomorphisms Similarity Similarity Invariants of Operators Inner Product Spaces Complex Vector Spaces Inner Products Orthogonality and Orthonormal Bases The Gram-Schmidt Process Unitary Matrices and Orthogonal Matrices Schur Factorization and the Cayley-Hamilton Theorem The QR Factorization and Applications Orthogonal Complements Projections Hermitian Matrices and Quadratic Forms Linear Functionals and the Adjoint of an Operator Hermitian Matrices Normal Matrices Quadratic Forms Singular Value Decomposition The Polar Decomposition Appendix A: Basics of Set Theory Appendix B: Summation and Product Notation Appendix C: Mathematical Induction Appendix D: Complex Numbers Answers/Hints to Odd-Numbered Problems Index A Summary appears at the end of each chapter.