Linear Algebra In Geology
Ilona Brownson
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!
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 $(AB)^-1 = B^-1A^-1$ when $A$ and $B$ are both invertible, one writes$$(AB) (B^-1A^-1) = A(BB^-1) A^-1 = A I A^-1 = A A^-1 = I.$$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?
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).
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.
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 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.)
Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. Its eigenvectors are those vectors that are only stretched, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or squished. If the eigenvalue is negative, the eigenvector's direction is reversed.[1]
The eigenvectors and eigenvalues of a linear transformation serve to characterize it, and so they play important roles in all the areas where linear algebra is applied, from geology to quantum mechanics. In particular, it is often the case that a system is represented by a linear transformation whose outputs are fed as inputs to the same transformation (feedback). In such an application, the largest eigenvalue is of particular importance, because it governs the long-term behavior of the system after many applications of the linear transformation, and the associated eigenvector is the steady state of the system.
There is a direct correspondence between n-by-n square matrices and linear transformations from an n-dimensional vector space into itself, given any basis of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices, or the language of linear transformations.[3][4]
Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German word eigen (cognate with the English word own) for 'proper', 'characteristic', 'own'.[6][7] Originally used to study principal axes of the rotational motion of rigid bodies, eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization.