Lectures in Abstract Algebra

1. Front Matter

   Pages i-xii

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2. Finite Dimensional Vector Spaces

   • Nathan Jacobson

   Pages 1-30

3. Linear Transformations

   • Nathan Jacobson

   Pages 31-62

4. The Theory of a Single Linear Transformation

   • Nathan Jacobson

   Pages 63-114

5. Sets of Linear Transformations

   • Nathan Jacobson

   Pages 115-135

6. Bilinear Forms

   • Nathan Jacobson

   Pages 136-171

7. Euclidean and Unitary Spaces

   • Nathan Jacobson

   Pages 172-198

8. Products of Vector Spaces

   • Nathan Jacobson

   Pages 199-226

9. The Ring of Linear Transformations

   • Nathan Jacobson

   Pages 227-237

10. Infinite Dimensional Vector Spaces

    • Nathan Jacobson

    Pages 238-276

11. Back Matter

    Pages 277-280

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The present volume is the second in the author's series of three dealing with abstract algebra. For an understanding of this volume a certain familiarity with the basic concepts treated in Volume I: groups, rings, fields, homomorphisms, is presup­posed. However, we have tried to make this account of linear algebra independent of a detailed knowledge of our first volume. References to specific results are given occasionally but some of the fundamental concepts needed have been treated again. In short, it is hoped that this volume can be read with complete understanding by any student who is mathematically sufficiently mature and who has a familiarity with the standard notions of modern algebra. Our point of view in the present volume is basically the abstract conceptual one. However, from time to time we have deviated somewhat from this. Occasionally formal calculational methods yield sharper results. Moreover, the results of linear algebra are not an end in themselves but are essential tools for use in other branches of mathematics and its applications. It is therefore useful to have at hand methods which are constructive and which can be applied in numerical problems. These methods sometimes necessitate a somewhat lengthier discussion but we have felt that their presentation is justified on the grounds indicated. A stu­ dent well versed in abstract algebra will undoubtedly observe short cuts. Some of these have been indicated in footnotes. We have included a large number of exercises in the text.

• Calculation
• Morphism
• algebra
• linear algebra
• mathematics
• matrix theory

• Department of Mathematics, Yale University, New Haven, USA

  Nathan Jacobson