Linear Algebra

This is a carefully written, clear account of elementary linear algebra with good problems and interesting historical sidelights. It will be used for collateral reading and primary reference by many students. The text, written by a very well-known author, includes many exercises and good examples.

Autorentext

Prof. Dr. Klaus Jänich, Regensburg

Inhalt

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 TheCharacteristic 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.