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I am writing a Linear Algebra textbook that is supposed to have several Earth Science applications (Geophysics, Atmospheric Science) given that I come from an Earth/Atmospheric Science background. I want to help students from my field to achieve a better understanding of Linear Algebra and how it is used in Earth Science. Examples include dynamical systems, principal component analysis, continuum mechanics, first-order Markov chains, signal processing with DFT, etc. I also want to retain some level of Mathematical rigors with detailed proofs, but I have found myself a bit long-winded about this. I have written some book proposals to large publishers but got rejected as they find the combination of Linear Algebra and Earth Science lacks market appeal. Despite this, I still want to finish the book even if I am on my own. Here I am asking for some suggestions regarding the content and coverage, strategies for any possible publishing opportunities, and how to promote the book itself. Particularly

  • How much/to what extent proofs are needed in the book: I understand that for Earth Science students proofs may not be necessarily what they need the most. However, as a Mathematics enthusiast, I can't stand giving out results without some degree of justification. I also hope that the readers who are interested in the theories can refer to the proofs in case they are looking for a deeper understanding of the materials. Also, how rigorous should they be: I am not from a pure Mathematics background so in some of my proofs I use heuristic arguments, or something along the line of "Without loss of generality/Similar to blah blah blah". I want the book to be used in courses but I am afraid if (Mathematics) teachers will find this problematic.

  • The balance between Mathematics and Earth Science applications: while my (expected) main audiences are from the fields of Earth Science, (Similar to the first question) I think I am spending a lot of time discussing the Mathematical theories. I admit that I want to hit two birds with one stone, but nevertheless I need more Earth Science applications to equalize. I want to know if any of you have suggestions about what potential applicational topics can be included.

  • Publishing or online free materials? I am interested in publishing my work, but as I mentioned I got rejected by large publishers (and while I understand their concern, I still think my book can be useful at least to some people). I have seen some people chose the path of independent publishing. I am also thinking if no publisher accept my proposal, I will just post my textbook online for free. In this case, I would like to ask whether it is appropriate to promote my Github repository online and invite people to give comments and open issues/requests to enrich the quality of the book. Specifically, I really hope to advertise my book so that more people will know about and use it, but I am quite nervous for that...

(I have written about 300 pages, that is, 40% of the expected content. I would like to know if I can post the link to my repository here so that you guys can take a look at it.) Thanks for the patience reading this question, any help and idea will be greatly appreciated!

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    $\begingroup$ This seems like a great project! Just a comment regarding "so in some of my proofs I use heuristic arguments, or something along the line of 'Without loss of generality [...]'": The phrase "without loss of generality" has a very specific meaning: it means "we prove the claim in a special case and the general case can be logically (and easily) reduced to this special case - hence our proof actually shows the claim in the general case!" Please do not use the phrase to indicate that you only show a special case (without the part that I set in bold) or that your argument is non-rigorous. $\endgroup$
    – Jochen Glueck
    Commented Jul 10 at 13:32
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    $\begingroup$ Needless to add, it's of course completely sensible in many cases - and in particular in applied textbooks - to give non-rigorous arguments or to prove only special cases (even if the general cases doesn't follow from them). I'm just advocating against indicating this with a phrase that actually means something else. $\endgroup$
    – Jochen Glueck
    Commented Jul 10 at 13:35
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    $\begingroup$ @JochenGlueck Thanks for the feedback! Sure, I will be more careful when using the phrase in the future. $\endgroup$
    – Benjamin_Gal
    Commented Jul 10 at 13:41
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    $\begingroup$ Whatever you decide. Be sure to save a copy of the 300 pages you've already written. Make a separate file then start your radical edit. Also, you might take a day to surf through some applied linear algebra texts. They'd give you more an idea about appropriate tone, detail and perhaps format. I have an applied linear algebra text written by Olver which is a masterpiece see www-users.cse.umn.edu/~olver/ala1.html (now it's a Springer UTM in its 2nd edition). Pdfs exist on university sites if you look. $\endgroup$
    – James S. Cook
    Commented Jul 11 at 4:12
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    $\begingroup$ You don't mention it so I'll note that your work would be a lot better if you had experience teaching students in your target audience, for at least a couple of semesters. You would find out, for example, whether proofs help (whether it would make the book more or less likely to be adopted is a different question, of course). $\endgroup$
    – Jim Hefferon
    Commented Jul 12 at 12:41

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Speaking as a pure mathematician: in a book like this, don't prove things for the sake of proving things.

You're writing for a niche market of students, so you have the luxury of being able to pin down what those students need to know, much more precisely than the author of a typical calc/diff-eq/linear algebra book. You don't need your book to be all things to all people, and there are plenty of rigorous Linear Algebra books that students can refer to for all the proofs.

Your test for whether to include a proof should be: does it help explain the concepts? That is, does it impart understanding to your intended audience more effectively than a non-rigorous explanation would?

Here's an example of a proof with good explanatory value. To prove that $(AB)^{-1} = B^{-1}A^{-1}$ when $A$ and $B$ are both invertible, one writes $$ (AB) (B^{-1}A^{-1}) = A(BB^{-1}) A^{-1} = A I A^{-1} = A A^{-1} = I. $$ This is short and very convincing. There's no reason not to include it.

With longer, detailed proofs, I can tell you exactly what will happen: students won't read them, and you'll have wasted a ton of time writing them. The only way to make students engage with long proofs is to test them on the substance of them, and if the teacher is doing that, it's a pure math course, so why would they use a book for earth scientists?

About heuristic arguments: it's fine to include them. Again, the test (as with anything) is whether they help impart understanding. But don't call them proofs if they aren't proofs.

There is a school of thought among mathematicians that any mathematics worth learning is worth learning in full rigor. Based on my experience teaching non-math students, I strongly disagree with this school of thought. One will sometimes hear a false dichotomy that the only choices for a math course are full rigor or rote memorization of problem-solving techniques, with no middle ground. This is false: the middle ground is to teach concepts. If you're not sure what I mean by concepts, I suggest reading Gilbert Strang's Introduction to Linear Algebra. It's not a particularly rigorous book, but it imparts a wealth of useful and even beautiful concepts. For that matter, anyone writing a book on Linear Algebra should look at Strang's book, because it's one of the best introductory books on the subject (with the only caveat being that it's not rigorous enough for pure math students).

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    $\begingroup$ Are you referring to Strang's Linear Algebra and Its Applications? (Sometimes confused with his Introduction to Linear Algebra.) $\endgroup$
    – J W
    Commented Jul 10 at 17:52
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    $\begingroup$ @JW Edited with the title I had in mind. I've never read Linear Algebra and its Applications. $\endgroup$
    – user1149748
    Commented Jul 10 at 18:05
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    $\begingroup$ Welp, it seems then I have to make a significant change to my work :/ $\endgroup$
    – Benjamin_Gal
    Commented Jul 10 at 18:29
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    $\begingroup$ This answer matches my experience as an engineering student. University was a long time ago, but two things stand out very clearly: a) I found proofs an impractically hard way to learn, and b) I used Strang's book to get through. It wasn't the course book, but it was the one I read in parallel with the course notes and text to learn and understand linear algebra. Interestingly, it was recommended by a prof who used a lot of proofs. Without Strang, I would've been just another "five-oh and go" (marginal pass) engineering math student. $\endgroup$
    – Jason
    Commented Jul 11 at 1:07
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    $\begingroup$ @Wastrel That's a valid choice in principle, but the work required to write such an appendix may be hard to justify given (1) the relatively small number of students who will read it (remember these are earth science students, not math students) (2) the wide availability of other Linear Algebra books that contain the proofs, and (3) the difficulty of bringing any textbook project to completion, especially without a publisher pushing you for chapters. $\endgroup$
    – user1149748
    Commented Jul 11 at 14:06
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This may sound a bit self-serving, but Noah McLean and I published a linear algebra book for Earth Scientists this spring through CRC. It is not proof-oriented but does try to appeal to geologists.

https://www.routledge.com/Linear-Algebra-for-Earth-Scientists/Walker-McLean/p/book/9781032555942

This is the description:
Linear Algebra for Earth Scientists is written for undergraduate and graduate students in Earth and Environmental sciences. It is intended to give students enough background in linear algebra to work with systems of equations and data in geology, hydrology, geophysics, or whatever part of the Earth Sciences they engage with.

The book does not presuppose any extensive prior knowledge of linear algebra. Instead, the book builds students up from a low base to a working understanding of the subject that they can apply to their work, using many familiar examples in the geosciences.

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    $\begingroup$ OP should certainly look at your book very carefully. There may or may not be room for two earth-science-focused linear algebra books, but even if there is, OP should get very clear about what his book offers that yours doesn't, before sinking more effort into his project. $\endgroup$
    – user1149748
    Commented Jul 12 at 17:50
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The answer by user1149748 is very good and covers several important points. I'd like to add a few further thoughts.

Other things to consider include what is your starting point and what do you assume the reader already knows well? Will you begin with vectors, matrices and solving linear systems of equations? How familiar should the reader be with calculus or differential equations? Also, what is your ending point? These days, many books reach the singular value decomposition, although some courses don't get that far. Some may not even reach eigenvalues and eigenvectors.

How about numerical algorithms? Do you want the course to give a taste of scientific computing? Will you use a programming language or other software, and if so which one(s)? Will you use library functions or go into the algorithms themselves? How much time, if any, will you spend on topics such as floating point numbers, convergence or how good approximations are?

Also, to what extent, if at all, will you introduce state estimation (Kalman filter), inverse problems or other topics that are often taught in a course of their own and have whole books devoted to them? Will you include optional topics chapters?

You might want to check out the recent Application-Inspired Linear Algebra by Moon, Asaki & Snipes. It centers on two main applications: diffusion welding and radiography. Perhaps a similar approach would work with two or three Earth Science applications?

I hope the above provides some food for thought. Whatever you put in, keep your main audience in mind and make it interesting and useful to them. Solve problems that they are likely to run into.

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  1. Looking at UT and Texas A&M geophysics majors, neither one requires a full course in linear algebra. Both have some LA as part of a capstone "math methods" course. I would be leery of the market for a full LA course.

https://catalog.utexas.edu/archive/2018-19/undergraduate/geosciences/degrees-and-programs/bs-geological-sciences/option-ii-geophysics/

https://catalog.tamu.edu/undergraduate/arts-and-sciences/geology-geophysics/geophysics-bs/#programrequirementstext

  1. In particular, I especially doubt the market for proof-oriented LA for geoscientists. You sort of even have to convince me they need the matrices!

  2. Note that geology students will be much more math-averse than geophysics.

  3. I would be a little wary of advanced, real uses of LA if the usage is too difficult (e.g. FT) for undergrad students. You might be better off with somewhat artificial problems that are easier matrix manipulation stuff and just use ideas from business or estimation aspects of mining and petroleum extraction and environmental topics. Think about how geoscientists (especially undergrads) use the tools they use. I bet it's a lot more like how organic chemists use NMR spectra, without needing to know the physics and math inside the machine. (But knowing a bunch of useful stuff about how to correlate spectra to molecules, why we use deuterated solvents, etc...that some physicist who knew a bunch of magnetism theory would totally miss.)

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