Introduction to Linear Algebra

1. Introduction to Linear Systems. 1.1. Linear systems: First Examples. 1.2. Matrices. 1.3. Matrices and Linear Systems. 1.4. The Gaussian Algorithm. 1.5. Exercises with Solutions. 1.6. Suggested Exercises. 2. Vector Spaces. 2.1. Introduction: The Set of Real Numbers. 2.2. The Vector Space Rn and the Vector Space of Matrices. 2.3. Vector Spaces. 2.4. SubSpaces. 2.5. Exercises with Solutions. 2.6. Suggested Exercises. 3. Linear Combination and Linear Independence. 3.1. Linear Combinations and Generators. 3.2. Linear Independence. 3.3. Exercises with Solutions. 3.4. Suggested Exercises. 4. Basis and Dimension. 4.1. Basis: Definition and Examples. 4.2. The Concept of Dimension. 4.3. Gaussian Algorithm. 4.4. Exercises with Solutions. 4.5. Suggested Exercises. 4.6. Appendix: The Completion Theorem. 5. Linear Transformations. 5.1. Linear Transformations: Definition. 5.2. Linear Maps and Matrices. 5.3. the Composition of Linear transformations. 5.4. Kernel and Image. 5.5. The Rank Nullity Theorem. 5.6. Isomorphism of Vector Spaces. 5.7. Calculation of Kernel and Image. 5.8. Exercises with Solutions. 5.9. Suggested Exercises. 6. Linear Systems. 6.1. Preimage. 6.2. Linear Systems. 6.3. Exercises with Solutions. 6.4. Suggested Exercises. 7. Determinant and Inverse. 7.1. Definition of Determinant. 7.2. Calculating the Determinant: Cases 2 _ 2 and 3 _ 3. 7.3. Calculating the Determinant with a Recursive Method. 7.4. Inverse of a Matrix. 7.5. Calculation of the Inverse with the Gaussian Algorithm. 7.6. The Linear Maps from Rn to Rn. 7.7. Exercises with Solutions. 7.8. Suggested Exercises. 7.9. Appendix. 8. Change of Basis. 8.1. Linear Transformations and Matrices. 8.2. The Identity Map. 8.3. Change of Basis for Linear Transformations. 8.4. Exercises with Solutions. 8.5. Suggested Exercises. 9. Eigenvalues and Eigenvectors. 9.1. Diagonalizability. 9.2. Eigenvalues and Eigenvectors. 9.3. Exercises with Solutions. 9.4. Suggested Exercises. 10. Scalar Products. 10.1. Bilinear Forms. 10.2. Bilinear Forms and Matrices. 10.3. Basis Change. 10.4. Scalar Products. 10.5. Orthogonal Subspaces. 10.6. Gram-Schmidt Algorithm. 10.7. Exercises with Solutions. 10.8. Suggested Exercises. 11. Spectral Theorem. 11.1. Orthogonal Linear Transformations. 11.2. Orthogonal Matrices. 11.3. Symmetric Linear Transformations. 11.4. The Spectral Theorem. 11.5. Exercises with Solutions. 11.6. Suggested Exercises. 11.7. Appendix: The Complex Case. 12. Applications of Spectral Theorem and Quadratic Forms. 12.1. Diagonalization of Scalar Products. 12.2. Quadratic Forms. 12.3. Quadratic Forms and in the Plane. 12.4. Exercises with Solutions. 12.5. Suggested Exercises. 13. Lines and Planes. 13.1. Points and Vectors in R3. 13.2. Scalar Product and Vector Product. 13.3. Lines in R3. 13.4. Planes in R3. 13.5. Exercises with Solutions. 13.6. Suggested Exercises. 14. Introduction to Modular Arithmetic. 14.1. The Principle of Induction. 14.2. The Division Algorithm and Euclid's Algorithm. 14.3. Congruence Classes. 14.4. Congruences. 14.5. Exercises with Solutions. 14.6. Suggested Exercises. 14.7. Appendix: Elementary Notions of Set Theory. Appendix A. Complex Numbers. A.1. Complex Numbers. A.2. Polar Representation. Appendix B. Solutions of Some Suggested Exercises. Bibliography. Index