1: Vectors in the Plane and Space. 2: Vector Spaces. 3: Examples of Vector Spaces. 4: Subspaces. 5: Linear Independence and Dependence. 6: Finite Dimensional Vector Spaces and Bases. 7: The Elements of Vector Spaces: A Summing Up. 8: Linear Transformations. 9: Linear Transformations: Examples and Applications. 10: Linear Transformations and Matrices. 11: Representing Linear Transformations by Matrices. 12: More on Representing Linear Transformations by Matrices. 13: Systems of Linear Equations. 14: The Elements of Eigenvalue and Eigenvector Theory. 15: Inner Product Spaces. 16: The Spectral Theorem and Quadratic Forms. 17: Jordan Canonical Form. 18: Application to Differential Equations. 19: The Similarity Problem. Appendix A: Multilinear Algebra and Determinants. B: Complex Numbers.