Linear Algebra

1. Vectors in the Plane and in Space.- 1.1 First Steps.- 1.2 Exercises.- 2. Vector Spaces.- 2.1 Axioms for Vector Spaces.- 2.2 Cartesian (or Euclidean) Spaces.- 2.3 Some Rules for Vector Algebra.- 2.4 Exercises.- 3. Examples of Vector Spaces.- 3.1 Three Basic Examples.- 3.2 Further Examples of Vector Spaces.- 3.3 Exercises.- 4. Subspaces.- 4.1 Basic Properties of Vector Subspaces.- 4.2 Examples of Subspaces.- 4.3 Exercises.- 5. Linear Independence and Dependence.- 5.1 Basic Definitions and Examples.- 5.2 Properties of Independent and Dependent Sets.- 5.3 Exercises.- 6. Finite-Dimensional Vector Spaces and Bases.- 6.1 Finite-Dimensional Vector Spaces.- 6.2 Properties of Bases.- 6.3 Using Bases.- 6.4 Exercises.- 7. The Elements of Vector Spaces: A Summing Up.- 7.1 Numerical Examples.- 7.2 Exercises.- 8. Linear Transformations.- 8.1 Definition of Linear Transformations.- 8.2 Examples of Linear Transformations.- 8.3 Properties of Linear Transformations.- 8.4 Images and Kernels of Linear Transformations.- 8.5 Some Fundamental Constructions.- 8.6 Isomorphism of Vector Spaces.- 8.7 Exercises.- 9. Linear Transformations: Examples and Applications.- 9.1 Numerical Examples.- 9.2 Some Applications.- 9.3 Exercises.- 10. Linear Transformations and Matrices.- 10.1 Linear Transformations and Matrices in IR3.- 10.2 Some Numerical Examples.- 10.3 Matrices and Their Algebra.- 10.4 Special Types of Matrices.- 10.5 Exercises.- 11. Representing Linear Transformations by Matrices.- 11.1 Representing a Linear Transformation by a Matrix.- 11.2 Basic Theorems.- 11.3 Change of Bases.- 11.4 Exercises.- 12. More on Representing Linear Transformations by Matrices.- 12.1 Projections.- 12.2 Nilpotent Transformations.- 12.3 Cyclic Transformations.- 12.4 Exercises.- 13. Systems of Linear Equations.- 13.1 Existence Theorems.- 13.2 Reduction to Echelon Form.- 13.3 The Simplex Method.- 13.4 Exercises.- 14. The Elements of Eigenvalue and Eigenvector Theory.- 14.1 The Rank of an Endomorphism.- 14.2 Eigenvalues and Eigenvectors.- 14.3 Determinants.- 14.4 The Characteristic Polynomial.- 14.5 Diagonalization Theorems.- 14.6 Exercises.- 15. Inner Product Spaces.- 15.1 Scalar Products.- 15.2 Inner Product Spaces.- 15.3 Isometries.- 15.4 The Riesz Representation Theorem.- 15.5 Legendre Polynomials.- 15.6 Exercises.- 16. The Spectral Theorem and Quadratic Forms.- 16.1 Self-Adjoint Transformations.- 16.2 The Spectral Theorem.- 16.3 The Principal Axis Theorem for Quadratic Forms.- 16.4 A Proof of the Spectral Theorem in the General Case.- 16.5 Exercises.- 17. Jordan Canonical Form.- 17.1 Invariant Subspaces.- 17.2 Nilpotent Transformations.- 17.3 The Jordan Normal Form.- 17.4 Square Roots.- 17.5 The Hamilton-Cayley Theorem.- 17.6 Inverses.- 17.7 Exercises.- 18. Application to Differential Equations.- 18.1 Linear Differential Systems: Basic Definitions.- 18.2 Diagonalizable Systems.- 18.3 Application of Jordan Form.- 18.4 Exercises.- 19. The Similarity Problem.- 19.1 The Fundamental Problem ofLinear Algebra.- 19.2 A Bit of Invariant Theory.- 19.3 Exercises.- A. Multilinear Algebra and Determinants.- A.1 Multilinear Forms.- A.2 Determinants.- A.3 Exercises.- B. Complex Numbers.- B.1 The Complex Numbers.- B.2 Exercises.- Font Usage.- Notations.