TY - BOOK AU - Janich, Klaus TI - Linear algebra SN - 81-8128-187-X PY - 2004/// CY - New Delhi PB - Springer India Ltd. KW - Mathematics KW - Algebras, Linear KW - Algebra N1 - 58 illustrations; 1. Sets and Maps -- 1.1 Sets -- 1.2 Maps -- 1.3 Test -- 1.4 Remarks on the Literature -- 1.5 Exercises -- 2. Vector Spaces -- 2.1 Real Vector Spaces -- 2.2 Complex Numbers and Complex Vector Spaces -- 2.3 Vector Subspaces -- 2.4 Test -- 2.5 Fields -- 2.6 What Are Vectors? -- 2.7 Complex Numbers 400 Years Ago -- 2.8 Remarks on the Literature -- 2.9 Exercises -- 3. Dimension -- 3.1 Linear Independence -- 3.2 The Concept of Dimension -- 3.3 Test -- 3.4 Proof of the Basis Extension Theorem and the Exchange Lemma -- 3.5 The Vector Product -- 3.6 The 'Steinitz Exchange Theorem' -- 3.7 Exercises -- 4. Linear Maps -- 4.1 Linear Maps -- 4.2 Matrices -- 4.3 Test -- 4.4 Quotient Spaces -- 4.5 Rotations and Reflections in the Plane -- 4.6 Historical Aside -- 4.7 Exercises -- 5. Matrix Calculus -- 5.1 Multiplication -- 5.2 The Rank of a Matrix -- 5.3 Elementary Transformations -- 5.4 Test -- 5.5 How Does One Invert a Matrix? -- 5.6 Rotations and Reflections (continued) -- 5.7 Historical Aside -- 5.8 Exercises -- 6. Determinants -- 6.1 Determinants -- 6.2 Determination of Determinants -- 6.3 The Determinant of the Transposed Matrix -- 6.4 Determinantal Formula for the Inverse Matrix -- 6.5 Determinants and Matrix Products -- 6.6 Test -- 6.7 Determinant of an Endomorphism -- 6.8 The Leibniz Formula -- 6.9 Historical Aside -- 6.10 Exercises -- 7. Systems of Linear Equations -- 7.1 Systems of Linear Equations -- 7.2 Cramer's Rule -- 7.3 Gaussian Elimination -- 7.4 Test -- 7.5 More on Systems of Linear Equations -- 7.6 Captured on Camera! -- 7.7 Historical Aside -- 7.8 Remarks on the Literature -- 7.9 Exercises -- 8. Euclidean Vector Spaces -- 8.1 Inner Products -- 8.2 Orthogonal Vectors -- 8.3 Orthogonal Maps -- 8.4 Groups -- 8.5 Test -- 8.6 Remarks on the Literature -- 8.7 Exercises -- 9. Eigenvalues -- 9.1 Eigenvalues and Eigenvectors -- 9.2 The Characteristic Polynomial -- 9.3 Test -- 9.4 Polynomials -- 9.5 Exercises -- 10. The Principal Axes Transformation -- 10.1 Self-Adjoint Endomorphisms -- 10.2 Symmetric Matrices -- 10.3 The Principal Axes Transformation for Self-Adjoint Endomorphisms -- 10.4 Test -- 10.5 Exercises -- 11. Classification of Matrices -- 11.1 What Is Meant by 'Classification'? -- 11.2 The Rank Theorem -- 11.3 The Jordan Normal Form -- 11.4 More on the Principal Axes Transformation -- 11.5 The Sylvester Inertia Theorem -- 11.6 Test -- 11.7 Exercises -- 12. Answers to the Tests -- References N2 - This book covers the material of an introductory course in linear algebra: sets and maps, vector spaces, bases, linear maps, matrices, determinants, systems of linear equations, Euclidean spaces, eigenvalues and eigenvectors, diagonalization of self-adjoint operators, and classification of matrices. The book is written for beginners. Its didactic features (the "book within a book" and multiple choice tests with commented answers) make it especially suitable for self-study ER -