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I'm searching for some material (books or lecture notes) that extensively uses a geometric approach to explain the meaning of the concepts related to vector spaces, matrices, and linear applications presented in an undergraduate course in linear algebra (for instance, the basis of a vector space, the orientation of a vector space, the determinant of a matrix, and so on). Nice pictures and graphics is a plus.

Martin Sleziak
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Dal
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    There is a certain amount of tension between rigor and geometric insight. "Rigorous" means amenable to algorithmic checking of the correctness of the logic. ${}\qquad{}$ – Michael Hardy Nov 21 '14 at 18:06
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    There are tons of textbooks doing exactly this, the mother of them all being Halmos' Finite dimensional vector spaces (1958). – Christian Blatter Nov 27 '14 at 12:55
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    i find strang's linear algebra textbook very helpful to see geometrically, for example i like the way he presents the four space associated with a linear transformation. – abel Nov 28 '14 at 17:19
  • I came in with Halmos's bookin my mind, and found it already stated. – Zhipu 'Wilson' Zhao Nov 29 '14 at 12:23

2 Answers2

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Two and a half hints from my side:

  • Linear Algebra from Klaus Jänich is an introductory text which presents a short and concise introduction to the most important concepts.

All books from Jänich written for undergraduates contain many clarifying pictures (I suppose his book Topology is the most famous of them). Some students may also appreciate his sometimes unconventional style (I was one of them).

Each chapter finishes with a section of exercises to check your understanding. A few sections are explicitly dedicated for mathematicians while other sections are written with focus for physicians.

But be aware, you probably will not find all the topics you need due to the restricted length of the book.

@Christian Blatter entitles this book the mother of modern books about Linear Algebra in the comment section and he's absolutely right. At the end of the Preface section Halmos gave credits to John von Neumann. He designates him as one of the originators of the modern spirit and methods which inspired him to present and teach the way he did in this book.

I deeply appreciate his books written from one of the great in a clear and easy to follow style. In order to get an impression of his writings I recommend his essay How to write mathematics.

  • The half hint is a supplement to Halmos book, namely his Linear Algebra Problem Book which provides completely elaborated solutions to about 160 problems. If you are interested in it you may have a look at this answer.
Martin Sleziak
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Markus Scheuer
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  • Once again, thank you. I've been already working through Halmos' Linear Algebra Problem Book and it has been a good reading. I will surely try the other two. – Dal Dec 03 '14 at 21:46
  • @Dal: Hi, Dal! Your response was amusing. I didn't recognise that you have posted the question. :-) Thanks for the bounty! Regards, – Markus Scheuer Dec 04 '14 at 07:54
  • It is curious that Fischer's very second rate textbook got translated. His presentation of the tensor product is so wrong – Mister Benjamin Dover Jan 11 '15 at 12:23
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Have a look a Choquet's "Neue Elementargeometrie" from 1970. There seem to be only French and German versions of the book, though.

Martin Sleziak
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Karsten W.
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