Linear Algebra

1. Front Matter

   Pages i-vii

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2. Vectors in the plane and space

   • Larry Smith

   Pages 1-12

3. Vector spaces

   • Larry Smith

   Pages 13-19

4. Subspaces

   • Larry Smith

   Pages 20-25

5. Examples of vector spaces

   • Larry Smith

   Pages 26-32

6. Linear independence and dependence

   • Larry Smith

   Pages 33-39

7. Bases and finite-dimensional vector spaces

   • Larry Smith

   Pages 40-53

8. The elements of vector spaces: a summing up

   • Larry Smith

   Pages 54-61

9. Linear transformations

   • Larry Smith

   Pages 62-86

10. Linear transformations: some numerical examples

    • Larry Smith

    Pages 87-98

11. Matrices and linear transformations

    • Larry Smith

    Pages 99-106

12. Matrices

    • Larry Smith

    Pages 107-124

13. Representing linear transformations by matrices

    • Larry Smith

    Pages 125-145

14. More on representing linear transformations by matrices

    • Larry Smith

    Pages 146-156

15. Systems of linear equations

    • Larry Smith

    Pages 157-182

16. The elements of eigenvalue and eigenvector theory

    • Larry Smith

    Pages 183-215

17. Inner product spaces

    • Larry Smith

    Pages 216-245

18. The spectral theorem and quadratic forms

    • Larry Smith

    Pages 246-273

19. Back Matter

    Pages 275-280

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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.

• Eigenvalue
• Eigenvector
• Lineare Algebra
• Vector space
• adopted textbook
• algebra
• calculus
• complex number
• linear algebra
• matrices
• quadratic form
• testing
• time
• transformation
• variable
• matrix theory

• Mathematisches Institut, Universität Göttingen, Göttingen, West Germany

  Larry Smith

• Department of Mathematics, Indiana University, Bloomington, USA

  Larry Smith

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