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 trans 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.
Inhaltsverzeichnis
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.
