This text is written for a course in linear algebra at the (U.S.) sophomore undergraduate level, preferably directly following a one-variable calculus course, so that linear algebra can be used in a course on multidimensional calculus. Realizing that students at this level have had little contact with complex numbers or abstract mathematics the book deals almost exclusively with real finite-dimensional vector spaces in a setting and formulation that permits easy generalization to abstract vector spaces. The parallel complex theory is developed in the exercises. The book has as a goal the principal axis theorem for real symmetric transformations, and a more or less direct path is followed. As a consequence there are many subjects that are not developed, and this is intentional. However a wide selection of examples of vector spaces and linear transยญ formations is developed, in the hope that they will serve as a testing ground for the theory. The book is meant as an introduction to linear algebra and the theory developed contains the essentials for this goal. Students with a need to learn more linear algebra can do so in a course in abstract algebra, which is the appropriate setting. Through this book they will be taken on an excursion to the algebraic/analytic zoo, and introduced to some of the animals for the first time. Further excursions can teach them more about the curious habits of some of these remarkable creatures
CONTENT
1 Vectors in the plane and space -- 2 Vector spaces -- 3 Subspaces -- 4 Examples of vector spaces -- 5 Linear independence and dependence -- 6 Bases and finite-dimensional vector spaces -- 7 The elements of vector spaces: a summing up -- 8 Linear transformations -- 9 Linear transformations: some numerical examples -- 10 Matrices and linear transformations -- 11 Matrices -- 12 Representing linear transformations by matrices -- 12bis 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
