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I realize that Axler's Linear Algebra Done Right is a great textbook, and the removal of determinants is often pedagogically sound. However, I was doing some problems on a practice entrance exam, and I was surprised by the amount of times solutions would involve having to apply determinants to geometric situations.

Therefore, my question is this: Where I can learn more about determinants, preferably without having to relearn all of linear algebra from another textbook?

John Clever
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  • Neil Strickland, Linear mathematics for applications (Appendix B and Section 12) does the most important proofs really well. From there I'd move on to Prasolov (Chapter I) and Krattenthaler. – darij grinberg Dec 16 '20 at 12:16
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    Actually, you can learn a lot here at this site, just search after "determinants mathematics stackexchange". For example, one of the hits for me was this nice post. I believe that there are several other interesting posts for you here, if you take the time to look them up. – Dietrich Burde Dec 16 '20 at 12:24
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    You can just keep reading Axler. Chapter 10, and specifically section 10B, is about the determinant. It will be much more productive to see the construction of the determinant "done right" and formally than go back to an introductory material now that you already have the Linear Algebraic baggage. Axler has some "applications" (if you want to call them that) of determinatns to volumes, for example, at the end of the chapter, and also connections with positive operators (which relate to calculus), etc. – Luiz Cordeiro Dec 16 '20 at 12:37

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The user who raised the question about three years ago might have learnt more about determinants. The reason for which I answer the question is that I want to give information that is probably too long for a comment.

There exists An introduction to determinants, a book about determinants for beginners (= persons who do not necessarily have a knowledge of linear algebra). This book is devoted to determinants, so there is little linear algebra in this book. Determinants in this book are defined recursively: for every square matrix $A$, $$ \det {(A)} = \begin{cases} [A]_{1,1}, & n = 1; \\ \displaystyle \sum_{i = 1}^{n} {(-1)^{i+1} [A]_{i,1} \det {(A(i|1))}}, & n \geq 2, \end{cases} $$ in which $[A]_{i,j}$ is the $(i,j)$-entry of $A$, and $A(i|j)$ the matrix obtained from $A$ by removing row $i$ and column $j$ (one can read this post for more information). It is also proved in this book that the Leibniz formula for determinants is true, so the author is not defining a new kind of determinants (one can read this post for more information).

It is hoped that anyone, espcially a person who does not bother to relearn all of linear algebra, can get started quickly, with this book.

A disadvantage is that the book is currently only in the Chinese language, but according to the author, the author will probably translate it into English when the author has much more spare time.

Juliamisto
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  • Any English version available somewhere? – Zack Fisher Aug 07 '24 at 22:34
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    @ZackFisher Sorry, there is no English version yet. – Juliamisto Aug 07 '24 at 22:34
  • Thanks, although this is surprising to me, because the Chinese text reads somewhat unnatural. I thought it was machine translated from a different language. – Zack Fisher Aug 07 '24 at 22:54
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    @ZackFisher I see. That is somewhat expected. Actually, I am the author, and I did not use machine translation. I have read more mathematical literature in English than in Chinese, so the Chinese text has been influenced by English. I hope that I will translate the book into English so that it can be read by more people around the world.... – Juliamisto Aug 07 '24 at 23:11
  • Sorry. I didn't intend to offend you. But it is actually easier to translate it into English because it is already written in a style closer to an English book. Great job! – Zack Fisher Aug 07 '24 at 23:23
  • @ZackFisher I agree with your remark. Actually, I was "translating my unwritten thoughts in English into Chinese" when writing this book. I am just too busy (and probably too lazy), because I am now a student who learns PDEs and dynamical systems as main subjects. So I said "if the author has much more spare time". (Incidentally, I am doing my homework about reaction–diffusion systems.) – Juliamisto Aug 07 '24 at 23:58
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For an easy introduction to determinants there are a lot of online resources. See for example

  1. Very basic intro to determinant

  2. Another intro to determinant with graphical representation

For a very nice graphic representation have a look at this youtube video.

Hope this helps!

Apprentice
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