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LITERATURE

There is no single set textbook for this course in linear algebra, but there are two recommended texts.

Klaus J¨anich’s Linear Algebra

This is published by Springer-Verlag in its “Undergraduate Texts in Mathematics” series.

It is the translation into English of one of the common textbooks for the first semester of first year at German universities. It is thoroughly modern in its approach. The author is a master expositor. He explains concepts with simplicity and great clarity, showing how linear algebra is used.

Charles W. Curtis’ Linear Algebra, An Introductory Approach

This is another book in Springer’s “Undergraduate Texts in Mathematics” series.

It is by an American author, and is written from a more traditional perspective than J¨anich’s book. It has different intentions, as the last few chapters show. It develops the theory further. The usefulness of linear algebra is illustrated in the last chapter, whose sections are devoted to applications in geometry, differential equations and number theory respectively.

There are many other books on the subject in most libraries. You should go to a library with a mathematics section and browse. Look for books with “Linear Algebra”, “Matrix Theory” and/or “Vector Spaces” in the title. In addition, most books on modern algebra or abstract algebra will contain at least a chapter on vector spaces or linear algebra.

You will find that some books will suit you, others will not. If you pursue mathematics further, you may well discover that some books which appeal to you now become less attractive later – and vice versa!

I would like to draw your attention to several books, at least one of which should suit your current tastes.

1. If you think you need a large number of worked examples, the Schaum’s Outline Series Volume on linear algebra is certainly full of them and you may find the book worth looking at.
2. One of the classics on the subject is Paul Halmos’ Finite Dimensional Vector Spaces. The author treats the subject from an “invariant” perspective: conceptual proofs are preferred over computational ones and the theory is developed as far as possible without appealing to explicit bases.
3. A more recent book by Paul Halmos is author his Linear Algebra Problem Book, which gives a very lucid exposition accompanied by a pertinent question in each section, together with (separate) hint and complete solution.
4. Another book which I can highly recommend you to look at is M. Hirsh and S. Smale Differential Equations, Dynamical Systems, and Linear Algebra. As the title suggests, the principal application the authors have in mind is to the theory of differential equations. Interestingly, this not only shows the unity of mathematics, illustrating the interconnection between topics, but also reflects the historical development.