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This note covers the following topics: Vector Spaces, Bases, Linear Maps, Matrices and Linear Maps, Direct Sums, Affine Maps, The Dual Space, Duality, Gaussian Elimination, LU, Cholesky, Echelon Form, Determinants, Vector Norms and Matrix Norms, Eigenvectors and Eigenvalues, Iterative Methods for Solving Linear Systems, Euclidean Spaces . This book features an ugly, elementary, and complete treatment of determinants early in . 2. By following a natural and unified approach to the subject it has, in less than 250 pages, achieved a more complete coverage of the subject than books with more than twice as many pages. Linear algebra is one of the most applicable areas of mathematics. This textbook is meant to be a mathematically complete and rigorous in-troduction to abstract linear algebra for undergraduates, possibly even first year students, specializing in mathematics. Some are peculiar to this book. 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