I have a B.S. in mathematics, statistics emphasis - and am currently in the second semester of Linear Models in a M.S. Statistics program.

Contrary to popular opinion, I don't think Linear Algebra Done Right is suitable for learning linear algebra. Statistics - as far as I've gathered - is more focused on what is called "numerical linear algebra," rather than the more algebraic (and more abstract) approach that Axler takes.

It took a lot of research on my part to find better books. I personally believe that these resources are much better for covering the linear algebra needed for linear models (I recommend these after a first-course treatment in linear algebra):

• Linear Algebra Done Wrong, Treil (funny title, hm?). I would recommend focusing on all of Ch. 1, all of Ch. 2 (skip 2.8), Ch. 3.1 through 3.5, all of Ch. 4, Ch. 5.1 through 5.4 (5.4 is extremely important). The only disadvantage of this book is that it isn't specifically geared toward statistics.
  
  
• Matrix Algebra by Gentle. Does not cover proofs, but it is a nice catalog of methods and ideas you should know for a stats program. Chapters 1 through 3 are essential material. Depending on the math prerequisites demanded, chapter 4 is nice to know. I would also recommend 5.8, 5.9, 6.7, 6.8, and 7.7. Chapters 8.2 - 8.5 are essential material, along with 9.1 - 9.2. This includes the linear model material as well that you will find in a M.S. program. All of the other stuff is optional or minimally covered in a stats program, as far as I know.
  
  
• Matrix Algebra From a Statistican's Perspective by Harville. This does not cover any of the linear model material itself, but rather the matrix algebra behind it. It is the most complete book I have found so far on linear algebra for statistics. For the most part, you should know Chapters 1 through 14, 16-18, 20, and 21.
  
  I have also heard that Matrix Algebra Useful for Statistics by Searle is good, but I haven't read it yet.
  
  If you feel like your linear algebra is particularly strong (i.e., you're comfortable with vector spaces, matrix operations, eigenvalues), you could try diving right into linear models. My personal favorite is Plane Answers to Complex Questions by Christensen. I reviewed this book on Amazon:
  
  >It's a decent text. If you want to understand any part of this text, you need to have at least a first course in linear algebra covering matrices and vector spaces, some probability, and some "mathematical maturity."
  
  >READ THE APPENDICES before you read any part of this text. READ THE APPENDICES. Take good notes on them and learn the appendices well. Then proceed to Chapter 1.
  
  >Definitely one of the most readable books I've read, but it does take a long time to digest everything. If you don't have a teacher to take you through this material and you're completely new to it, you will find that some details are omitted, but these details aren't complicated enough that someone with an undergraduate degree in math wouldn't be able to figure them out.
  
  >Highly recommended. The only thing I don't like about this text is some of its notation. It uses Cov(A) to mean the variance-covariance matrix of a random vector A, and Cov(A, B) to mean E[(A-E[A])(B-E[B])^transpose ]. I prefer using Var(A) for the former case. Furthermore, it uses ' instead of T to denote the transpose of a matrix.
  
  No linear models text will cover all of the linear algebra used, however. If you get a linear models text, you should get your hands on one of the above linear algebra texts as well.
  
  If you need a first course's treatment in Linear Algebra, I prefer [Linear Algebra and Its Applications](http://www.amazon.com/Linear-Algebra-Its-Applications-Edition/dp/0201709708) by Lay. The 3rd edition will suffice, although I think it's in the 5th edition now. Larson's [Elementary Linear Algebra*](http://www.amazon.com/Elementary-Linear-Algebra-Ron-Larson/dp/1133110878/ref=sr_1_1?s=books&ie=UTF8&qid=1458047961&sr=1-1&keywords=larson+linear+algebra) is also a decent text; older editions are likely cheaper, but will likely give you a similar treatment as well, so you may want to look into these too. I learned from the 6th edition in my undergrad.

I'm no physicist. My degree is in computer science, but I'm in a somewhat similar boat. I read all these pop-science books that got me pumped (same ones you've read), so I decided to actually dive into the math.

​

Luckily I already had training in electromagnetics and calculus, differential equations, and linear algebra so I was not going in totally blind, though tbh i had forgotten most of it by the time I had this itch.

​

I've been at it for about a year now and I'm still nowhere close to where I want to be, but I'll share the books I've read and recommend them:

• First and foremost, read Feynman's Lectures on Physics and do not skip a lecture. You can find them free on the link there, but they also sell the 3 volumes on amazon. I love annotating so I got myself physical copies. These are the most comprehensible lectures on anything I've ever read. Feynman does an excellent job on teaching you pretty much all of physics + math (especially electromagnetics) up until basics of Quantum Mechanics and some Quantum Field Theory assuming little mathematics background.
  
• Feyman lectures on Quantum Electrodynamics (The first Quantum Field Theory). This is pop-sciency and not math heavy at all, but it provides a good intuition in preparation for the bullet points below
  
• You're going to need Calculus. So if you're not familiar comfortable with integral concepts like integration by parts, Quantum Mechanics will be very difficult.
  
• I watched MIT's opencourseware online lectures on Quantum Mechanics and I did all the assignments. This gave me what I believe is a solid mathematical understanding on Quantum Mechanics
  
• I'm currently reading and performing exercises from this Introduction to Classical Field Theory. . This is just Lagrangian Field Theory, which is the classical analog of QFT. I'm doing this in preparation for the next bullet-point:
  
• Quantum Field Theory in a Nutshell. Very math heavy - but thats what we're after isnt it? I havent started on this yet since it relies on the previous PDF, but it was recommended in Feynmans QED book.
  
• I've had training on Linear Algebra during my CS education. You're going to need it as well. I recommend watching this linear algebra playlist by 3Blue1Brown. It's almost substitute for the rigorous math. My life would've been a lot easier if that playlist existed before i took my linear algebra course, which was taught through this book.
  
• Linear Algebra Part 2 - Tensor analysis! You need this for General Relativity. This is the pdf im currently reading and doing all the exercises. This pdf is preparing me for...
  
• Gravity. This 1000+ page behemoth comes highly recommended by pretty much all physicist I talk to and I can't wait for it.
  
• Concurrently I'm also reading this book which introduces you to the Standard Model.
  
  ​
  
  I'm available if you want to PM me directly. I love talking to others about this stuff.

Entrepreneur Reading List

1. Disrupted: My Misadventure in the Start-Up Bubble
   
2. The Phoenix Project: A Novel about IT, DevOps, and Helping Your Business Win
   
3. The E-Myth Revisited: Why Most Small Businesses Don't Work and What to Do About It
   
4. The Art of the Start: The Time-Tested, Battle-Hardened Guide for Anyone Starting Anything
   
5. The Four Steps to the Epiphany: Successful Strategies for Products that Win
   
6. Permission Marketing: Turning Strangers into Friends and Friends into Customers
   
7. Ikigai
   
8. Reality Check: The Irreverent Guide to Outsmarting, Outmanaging, and Outmarketing Your Competition
   
9. Bootstrap: Lessons Learned Building a Successful Company from Scratch
   
10. The Marketing Gurus: Lessons from the Best Marketing Books of All Time
    
11. Content Rich: Writing Your Way to Wealth on the Web
    
12. The Web Startup Success Guide
    
13. The Best of Guerrilla Marketing: Guerrilla Marketing Remix
    
14. From Program to Product: Turning Your Code into a Saleable Product
    
15. This Little Program Went to Market: Create, Deploy, Distribute, Market, and Sell Software and More on the Internet at Little or No Cost to You
    
16. The Secrets of Consulting: A Guide to Giving and Getting Advice Successfully
    
17. The Innovator's Solution: Creating and Sustaining Successful Growth
    
18. Startups Open Sourced: Stories to Inspire and Educate
    
19. In Search of Stupidity: Over Twenty Years of High Tech Marketing Disasters
    
20. Do More Faster: TechStars Lessons to Accelerate Your Startup
    
21. Content Rules: How to Create Killer Blogs, Podcasts, Videos, Ebooks, Webinars (and More) That Engage Customers and Ignite Your Business
    
22. Maximum Achievement: Strategies and Skills That Will Unlock Your Hidden Powers to Succeed
    
23. Founders at Work: Stories of Startups' Early Days
    
24. Blue Ocean Strategy: How to Create Uncontested Market Space and Make Competition Irrelevant
    
25. Eric Sink on the Business of Software
    
26. Words that Sell: More than 6000 Entries to Help You Promote Your Products, Services, and Ideas
    
27. Anything You Want
    
28. Crossing the Chasm: Marketing and Selling High-Tech Products to Mainstream Customers
    
29. The Innovator's Dilemma: The Revolutionary Book that Will Change the Way You Do Business
    
30. Tao Te Ching
    
31. Philip & Alex's Guide to Web Publishing
    
32. The Tao of Programming
    
33. Zen and the Art of Motorcycle Maintenance: An Inquiry into Values
    
34. The Inmates Are Running the Asylum: Why High Tech Products Drive Us Crazy and How to Restore the Sanity
    
    

    Computer Science Grad School Reading List
    

    
    

35. All the Mathematics You Missed: But Need to Know for Graduate School
    
36. Introductory Linear Algebra: An Applied First Course
    
37. Introduction to Probability
    
38. The Structure of Scientific Revolutions
    
39. Science in Action: How to Follow Scientists and Engineers Through Society
    
40. Proofs and Refutations: The Logic of Mathematical Discovery
    
41. What Is This Thing Called Science?
    
42. The Art of Computer Programming
    
43. The Little Schemer
    
44. The Seasoned Schemer
    
45. Data Structures Using C and C++
    
46. Algorithms + Data Structures = Programs
    
47. Structure and Interpretation of Computer Programs
    
48. Concepts, Techniques, and Models of Computer Programming
    
49. How to Design Programs: An Introduction to Programming and Computing
    
50. A Science of Operations: Machines, Logic and the Invention of Programming
    
51. Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology
    
52. The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation
    
53. The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine
    
54. Computability: An Introduction to Recursive Function Theory
    
55. How To Solve It: A New Aspect of Mathematical Method
    
56. Types and Programming Languages
    
57. Computer Algebra and Symbolic Computation: Elementary Algorithms
    
58. Computer Algebra and Symbolic Computation: Mathematical Methods
    
59. Commonsense Reasoning
    
60. Using Language
    
61. Computer Vision
    
62. Alice's Adventures in Wonderland
    
63. Gödel, Escher, Bach: An Eternal Golden Braid
    
    

    Video Game Development Reading List
    

    
    

64. Game Programming Gems - 1 2 3 4 5 6 7
    
65. AI Game Programming Wisdom - 1 2 3 4
    
66. Making Games with Python and Pygame
    
67. Invent Your Own Computer Games With Python
    
68. Bit by Bit

Sorry for a late reply. I've never read Jones' book. The selection of topics looks great, but the Amazon reviews are worrisome (lack or rigor, not self-contained).

From the preface of Rotman

>By the end of the nineteenth century, there were two main streams of group theory: topological groups (especially Lie groups) and finite groups... It is customary, nowadays, to approach our subject by two paths: "pure" group theory (for want of a better name) and representation theory.

From my experience, on the one hand, physicists don't need to know a lot of topics in "pure" group theory like the proof of structure theorem of finitely generated abelian groups or Sylow theorems. On the other hand, sometimes it's not clear when physicists talk about a group if they're actually talking about its Lie algebra (if it's a Lie group) or their representations. You can come out of a class on Lie groups without having learned much about Lie groups at all (like me)!

So if I have to recommend, the middle way would be to focus on representation theory, starting on finite groups first. Mikhail Khovanov listed a bunch of resources for his course http://www.math.columbia.edu/~khovanov/finite/ including Serre and Etingof. The first third of Serre is good. (It was written for chemists.) And my abstract algebra professor recommended Etingof. (It was from a course for bright high school students and another course for undergrads.)

A lot of results for finite groups carry over to compact infinite groups (as Serre explains). For Lie groups, it's more convenient because they have Lie algebras. Physicists also like the Lie algebra approach because they can bypass differential geometry. (Lee's Smooth Manifolds book also has a few chapters on Lie groups.) Two short books that I like are Cahn and Carter et al.

This is the level that I'm at. Beyond this you will have to look around and see which path you want to take. Hall and Fulton & Harris teach through examples but don't offer a uniform view of the subjects. I begin to like Kirillov Jr. and Vinberg & Onishchik. Procesi is inspiring to me because invariant theory like Schur-Weyl duality are actually useful in quantum information.

Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.

Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.

Intro to Math:

1. Book of Proof by Richard Hammack. This free book will show/teach you how mathematicians think. There are other such books out there. For example,
   
   

• How to Prove It: A Structured Approach by Daniel Velleman
  
  
• Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand et al
  
  
• Discrete Mathematics with Applications by Susanna Epp
  
  These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
  
  Intro to Abstract Algebra:
  
  
• Linear Algebra: Step by Step by Singh
  
  
• Abstract Algebra: A Student-Friendly Approach by the Dos Reis
  
  
• Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
  
  
• Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
  
  
• A Friendly Introduction to Group Theory by Nash
  
  
• Linear Algebra Done Right by Sheldon Axler
  
  There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
  
  
• Algebra: Chapter 0 by Aluffi
  
  
• Algebra by Maclane/Birkhoff
  
  
• Algebra by Lang
  
  
• Advanced Linear Algebra by Steven Roman
  
  Intro to Real Analysis:
  
  
• The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg
  
  
• Understanding Analysis by Abbot
  
  
• Writing Proofs in Analysis by Kane
  
  
• The Lebesgue Integral for Undergraduates by Johnston
  
  Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
  
  Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
  
  
• Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by the Hubbards
  
  At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
  
  From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
  
  Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
  
  Good luck.

I was in the same position as you in high school (and am finishing my math major this semester). Calculus is not "math" in the sense you're referring to it, which is pure mathematics, without application, just theory and logic. Calculus, as it is taught in high school, is taught as a tool, not as a theory. It is boring, tedious, and has no aesthetic appeal because it is largely taught as rote memorization.

Don't let this bad experience kill your enthusiasm. I'm not sure what specifically to recommend to you to perk your enthusiasm, but what I did in high school was just click around Wikipedia entries. A lot of them are written in layman enough terms to give you a glimpse and you inspire your interest. For example, I remember being intrigued by the Fibonacci series and how, regardless of the starting terms, the ratio between the (n-1)th and nth terms approaches the golden ratio; maybe look at the proof of that to get an idea of what math is beyond high school calculus. I remember the Riemann hypothesis was something that intrigued me, as well as Fermat's Last Theorem, which was finally proved in the 90s by Andrew Wiles (~350 years after Fermat suggested the theorem). (Note: you won't be able to understand the math behind either, but, again, you can get a glimpse of what math is and find a direction you'd like to work in).

Another thing that I wish someone had told me when I was in your position is that there is a lot of legwork to do before you start reaching the level of mathematics that is truly aesthetically appealing. Mathematics, being purely based on logic, requires very stringent fundamental definitions and techniques to be developed first, and early. Take a look at axiomatic set theory as an example of this. Axiomatic set theory may bore you, or it may become one of your interests. The concept and definition of a set is the foundation for mathematics, but even something that seems as simple as this (at first glance) is difficult to do. Take a look at Russell's paradox. Incidentally, that is another subject that captured my interest before college. (Another is Godel's incompleteness theorem, again, beyond your or my understanding at the moment, but so interesting!)

In brief, accept that math is taught terribly in high school, grunt through the semester, and try to read farther ahead, on your own time, to kindle further interest.

As an undergrad, I don't believe I yet have the hindsight to recommend good books for an aspiring math major (there are plenty of more knowledgeable and experienced Redditors who could do that for you), but here is a list of topics that are required for my undergrad math degree, with links to the books that my school uses:

• elementary real analysis
  
• linear algebra
  
• differential equations
  
• abstract algebra
  
  And a couple electives:
  
  
• topology
  
• graph theory
  
  And a couple books I invested in that are more advanced than the undergrad level, which I am working through and enjoy:
  
  
• abstract algebra
  
• topology
  
  Lastly, if you don't want to spend hundreds of dollars on books that you might not end up using in college, take a look at Dover publications (just search "Dover" on Amazon). They tend to publish good books in paperback for very cheap ($5-$20, sometimes up to $40 but not often) that I read on my own time while trying to bear high school calculus. They are still on my shelf and still get use.

I've structured my answer in 3 parts: how (abstract), why and how (now, concrete).

---

How do I math?

Practice. Math is a skill that you learn. You learn math much like you learn a second language, or the way artists get better at shadowing. Some people are fascinated by mathematics in and of itself, but you can just as well consider it a tool. In either case, fluency is the result of practice.

Obviously I'm not talking about learning equations by rote, the way you might learn a foreign vocabulary. Practice is perseverance. You're flexing your brain in a way that it's not used to, and that's tiring. Some people think they're bad at math because they're staring in a daze at a set of equations, with a slowly building headache. This doesn't mean that "math isn't for you," or any such bullshit. You should liken it to the way your muscles start hurting when you're working out. Hell, consider it a rite of passage. Because that's the good news: it gets better. You can get fluent in handling the tools you've struggled to learn, and each new tool exponentially increases the number of problems you're able to solve.

---

Why do I math? And why do you math so strange?

Each new tool exponentially increases the number of problems you're able to solve. This expresses itself in a number of different ways:

1. New problem classes. Some systems are non-linear and require differential equations. Some systems don't have exact solutions and require numerical mathematics. etc. These all require special tools.
   
   
2. Expressivity. With a bigger tool box, it becomes easier to express properties about a system, or to capture information in an abstract model. Imagine if you had to write a dissertation using only the 1000 most common words in the English language (similar to XKCD's project). Hard, huh? You're basically moving some of the work up-front; struggling to learn these new concepts now makes it easier to communicate later on.
   
   
3. Alternative paradigms. This is the least tangible, but one of the most important ones. This also ties into the roundabout methods they sometimes use. You can often attack any abstract problem from many different angles. These paradigm shifts are something that also only come from lots of practice. You're 100% right that there's no immediate gain in solving a problem in a roundabout way, but...
   
   a. Flexibility. You won't always see, or be able to solve those problems in a straightforward way. One of my high-school teachers once told me that "[I] went from Boston to New York via Paris". I told her she was right (I couldn't see how to solve the problem, so I took a bit of a detour), but that I did end up in New York.
   
   b. Easier to expand. There's a difference between solving a problem (in which case you want the most efficient technique) or understanding how a problem might be solved (what solution strategies exist and most crucially why they work/"the idea"). By being able to look at a problem in different ways, you gain additional understanding. You might be looking at a new mathematical concept and say, "hey, this is really just X in a different way."
   
   tl;dr: Math is many tools. You won't always have the best one for the job. You need to build a framework of understanding: you'll want to see the relation between different solution strategies and why they work. By knowing these relations, you'll be able to attack problems from multiple fronts. This ability makes you a better scientist.
   
   ---
   
   

   So HOW do I math?
   

   
   Your question shows you have the will. What you're looking for is a way. Not knowing your background, I don't know how good you are at learning stuff, at what level you're working (HS/Uni), etc. but I can give you a few tips that come from my own experience. Your mileage may vary; I'm no authority on education.
   
   

4. It doesn't matter how you learn. Most programs require you to pay attention to a lecturer. Personally, I suck at this. Maybe you do to. It doesn't matter! If you're better at getting your information from books, read^(†)! If you're better with videos, watch^(††)! Find the way that works for you. For me, it's with a book, alone in my room. I don't go to any lectures anymore, but my grades are great! Other people I know are exactly the other way around. There's no one-size-fits-all.
   
   
5. Practice. Yes, we went over this, but seriously. Try to do theory + exercises in parallel. It cements your understanding and makes math more fun (once you reach a certain fluency, solving new problems isn't much different from solving, say, sudokus).
   
   †: Look to the reference material of your course first. If you're at HS level I have no particular recommendations for you, but you could definitely get your fix from one of the math or teaching subreddits. For Uni level, Pearson books are really good for most sciences. I'm a big fan in particular (non-Pearson) of Lay's Linear Algebra and its Applications for (you guessed it!) linear algebra.
   
   ††: Kahn's academy is great for HS level. MIT open courseware (or the equivalent thereof for Harvard, etc.) is great for uni-level stuff. Being able to change the playback speed, pause & rewind makes all the difference for me.

I learned a lot from getting a copy of Rudin (however, this book is very challenging and probably not the best to self study from. I was able to get to about continuity before taking my analysis course and it was challenging, but worth while). You can probably find it online somewhere for free.

A teacher lent Introduction to Analysis to me and suggested I use it instead of the book by Rudin. It was a well written book and had exercises which were much more approachable (although it included very difficult ones as well). The layout of this book (and I'd bet many others) is quite similar to that of Rudin. It was nice to be able to read them together.

For linear algebra, I can't speak to the quality of many books, but there are plenty which can fairly easily be found online. You will likely be recommended Linear Algebra Done Right however I found it a bit challenging as a first introduction to linear algebra and never got quite far.

My university course used Larson, Falvo Linear Algebra and it was enjoyable and helps you learn the computations very well and gives a decent understanding of proofs.

I do highly recommend Genome by Matt Ridley and A History of God by Karen Armstrong. It looks like Before the Big Bang might be a great idea too.

However, I'm noticing a bit of redundancy in your stacks and don't want you to get bored! In the presence of the other books, I would recommend Dawkins' The Ancestor's Tale in lieu of The Greatest Show on Earth. (Although, if you're actually not going to read all the other books, I would actually go the other way.) Similarly, I would probably choose either to read the God Delusion or a few of the other books there.

Other recommendations: how about The Red Queen by Matt Ridley, and The Seven Daughters of Eve by Bryan Sykes? These occupy niches not covered by the others.

The popular expositions on cosmology all look supremely awesome, but you should probably choose half of them. Another idea: read just The Fabric of the Cosmos by Greene, and if you love it, go ahead and learn mechanics, vector calculus, Electrodynamics, linear algebra, and Quantum Mechanics! Hmm...on second thought, that might actually take longer than just reading those books :)

Since you are a practicing engieer with plenty of experience, I will suggest the right way to learn rather than the speed-of-the-internet , show-me-a-web-page way to acquire jargon.

Buy and read textbooks.

Start with Numerical Recipes by Press et al (Link 1). It has a couple of chapters on optimization and some very VERY excellent discussion. It will teach you the way academics formulate these problems, and how they solve them today.

Then read Gill, Murray, and Wright "Practical Optimization" (Link 2).

Next comes Roger Fletcher, "Practical Methods of Optimization". This book has been published two different ways: as a single volume, and also split into two volumes. Since Amazon Used Books sells the two volumes for considerably less money, I recommend that path: (Link 3) and (Link 4) .

After you have read those books, you will be able to appreciate the following paragraph:

I myself have found, in practice, that some of the old 1960's approaches to optimization work DELIGHTFULLY WELL on 2015 real world engineering problems, using 2015 computer power. In fifty years the problems have become 10,000 times more difficult and the computers have become 2^(50/3) times more powerful. The computers are winning the tug of war.

Make an honest try to solve your problem using no-derivative unconstrained optimizers, plus penalty functions or barrier functions for the constraints. I think you will be very pleasantly surprised. If you have honestly done your best and tried your hardest to get this to work, and failed, then your fallback is to implement the full stochastic miasma. Start with the TOMS paper by Corana, Marchesi, Martini, and Ridella. It is the most engineering-results oriented discussion I know of. If you are a masochist, try (just try!) to read the various publications and white papers by Lester Ingber. You will regret it.

Theory of Continuous Groups by Loewner. This book is based on lecture notes which Loewner was planning to turn into a larger book. Unfortunately he passed away before getting much done so some of his colleagues edited and compiled the notes into this book. I'm only quarter of the way in but so far it's given me a really unique perspective into group actions. I'm loving it but it doesn't hold my attention for long spans of time.

Geometry of Polynomials by Marden. Marden is my idol, and I plan to devote my life to studying the zeros of functions. That said, this book is the hardest goddamn book I have ever read. Hell, some of the exercises he gives were actual topics of published research 60 years ago. That seems a little mean to me. Anyway I still love this shit.

Mr. Tompkins in Paperback by Gamow. Alternates between stories about a character transplanted into hypothetical worlds where particular laws of physics are exaggerated and semi-rigorous lectures about the physics itself. The section on gravity as curvature of space was especially enlightening. The author uses the idea of a merry-go-round spinning at relativistic speed, so that straight lines on the surface (i.e. geodesics) are in fact curved to outside observers. You can then imagine that the merry-go-round is walled off from the outside, so that on the inside the centrifugal force can be thought of as gravity toward the edge. This is the concept of acceleration of reference frame being equivalent to gravity. For a non-physicist this kind of explanation is AWESOME.

Stranger in a Strange Land by Heinlein. My first Heinlein, just started it but I'm enjoying it so far. I honestly confused him with Haldeman... I loved The Forever War and I wanted to get another book by the author. Oh well.

Yeah so what I'm a nerd.

Intro Calculus, in American sense, could as well be renamed "Physics 101" or some such since it's not a very mathematical course. Since Intro Calculus won't teach you how to think you're gonna need a book like How to Solve Word Problems in Calculus by Eugene Don and Benay Don pretty soon.

Aside from that, try these:

Excursions In Calculus by Robert Young.

Calculus:A Liberal Art by William McGowen Priestley.

Calculus for the Ambitious by T. W. KORNER.

Calculus: Concepts and Methods by Ken Binmore and Joan Davies

You can also start with "Calculus proper" = Analysis. The Bible of not-quite-analysis is:

[Calculus by Michael Spivak] (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1413311074&sr=1-1&keywords=spivak+calculus).

Also, Analysis is all about inequalities as opposed to Algebra(identities), so you want to be familiar with them:

Introduction to Inequalities by Edwin F. Beckenbach, R. Bellman.

Analytic Inequalities by Nicholas D. Kazarinoff.

As for Linear Algebra, this subject is all over the place. There is about a million books of all levels written every year on this subject, many of which is trash.

My plan would go like this:

1. Learn the geometry of LA and how to prove things in LA:

Linear Algebra Through Geometry by Thomas Banchoff and John Wermer.

Linear Algebra, Third Edition: Algorithms, Applications, and Techniques
by Richard Bronson and Gabriel B. Costa.

2. Getting a bit more sophisticated:

Linear Algebra Done Right by Sheldon Axler.

Linear Algebra: An Introduction to Abstract Mathematics by Robert J. Valenza.

Linear Algebra Done Wrong by Sergei Treil.

3. Turn into the LinAl's 1% :)

Advanced Linear Algebra by Steven Roman.

Good Luck.

Thank you! This is exactly what I was looking for!!!! I didn't think anyone was going to give me a sufficient reply because there are a lot of books (sorry), but this is what I wanted. Where would you place the two books I linked, Principles and Techniques in Combinatorics and Introduction to Combinatorial Mathematics, Liu, in that list or would you consider studying them a redundant exercise? I also did not include this book in the list, but where would you place Problems from the Book and its accompanying Straight from the Book?

I will likely end up replacing the Graph Theory book I have in the list, by Berge, with Modern Graph Theory by Bellobas, since Berge doesn't have exercises, but I will assume it stays in the same order of the sequence.


I apologize for not initially including them. I did not realize that I did not. Also, are there any other topics you would recommend I cover for establishing a solid foundation. I didn't buy Rudin's Complex Analysis because I didn't know if that kind of thing was necessary. I don't even know what other branches of mathematics Complex Analysis relates to. There could be other topics I'm not aware of as well. Please don't hesitate to make more recommendations. I appreciate it.

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

you're going to struggle with mathematics until you get a better handle on stuff like proofs. it gets a little better once you've paid your dues.

book wise i'd recommend Linear Algebra and Its Applications by Peter Lax really just because i'm a huge fan of his. additionally, i'd recommend reading (or trying to read) every book you can get your hands on.

This is how I taught myself http://math.stanford.edu/~vakil/725/course.html

This book is a great help too http://www.amazon.com/Introduction-Homological-Cambridge-Advanced-Mathematics/dp/0521559871

Good luck, there is A LOT for you to learn.

Here you go:

> Calculus is the foundation for modern math. Always a good thing to have.

> Linear Algebra is the foundation for 3d mathematics in games. Things such as perspective projection, arbitrary rotation, and more exotic things such as quaternions come around here.

> Essential Math for Games is a most excellent book that, after having built your mathematical foundation through the previous two books, provides you pretty much everything you need to know about making a 3d renderer, which is probably one of the most educational experiences that you can undergo in game development.

I think Linear Algebra by Kuldeep Singh is the best fit for newcomers to LA. It's unpretentious and meant to be actually read by students (can you imagine?). This book will take you from someone who just discovered there exists such a thing as LA to someone who solves problems in Linear Algebra Done Right By Axler cold. After Kuldeep Singh you can pick up Advanced Linear Algebra by Steven Roman which is an extreme overkill even for mathematicians.

Basically, once you get the basics of LA down, you can simply read up on the newest matrix algos for machine learning on ArXiv or something. BTW, if your goal is working with data you need to learn some probability.

Somehow no one mentioned it (also in the referred math.stackexchange), but from abstract mathematical point of view, this is an awesome book IMO:

http://www.amazon.com/Multilinear-Algebra-Universitext-Werner-Greub/dp/0387902848

Linear Algebra and its applications.

http://www.amazon.com/Linear-Algebra-Its-Applications-3rd/dp/0201709708

is a great introductory text. lots of examples, lots of diagrams to illustrate important concepts. it'll take you from zero to the singular value decomposition.

i have a softcover version but couldn't find it on amazon.

Transistor-level IC designer.

The elective that benefited me the most was "Minimization of Functions" in the Applied Math department. This is a course in nonlinear optimization, where you learn how to find (numerically) the maxima and minima of highly nonlinear functions. It has been incredibly useful throughout my career, even in the simplest cases a/k/a curve-fitting.

The follow-on course, "Minimization of Functionals" was for the aero/astro Optimal Control types, people who want to shoot down ballistic missiles using ballistic missiles -- although the course catalog expressed it slightly differently. I skipped that one.

The book by Gill, Murray, and Wright gives a wonderful overview of the field (amazon link) but I recommend you buy a used copy or a previous-edition used copy. There ain't much that's changed in 40 years, except the cost of a trillion floating point operations has fallen by a factor of a million.

Many people like Strang's book for a standard treatment. Then getting a bit more general is Axler's Linear Algebra Done Right. But for further understanding, especially for computer graphics I recommend learning about the exterior and geometric algebras.

https://www.amazon.com/Linear-Algebra-via-Exterior-Products/dp/140929496X/

https://www.amazon.com/Linear-Geometric-Algebra-Alan-Macdonald/dp/1453854932

I dunno about “undergraduate”, but you could try Birkhoff & Mac Lane or Greub. Those are both kind of old, so someone else may have a better idea.

(co)Homology is useful whenever you have a ring and a module. Homological Algebra is, very roughly, linear algebra for a general ring over a module. (co)Homology problems turn up when you do something "linear" to an exact sequence of modules and they don't remain exact. Some examples that come to mind are:

Group Theory: Group Cohomology via Peter Webb's notes: The second cohomology group classifies certain kinds of group extensions.

Number Theory: Galois Cohomology a flavor of Group Coho, Class Field Theory which can be thought of, among other interpretations, as a vast extension of quadratic reciprocity,

Commutative Algebra: e.g. Local Cohomology ala Huneke, Gorenstein Rings, see page 6 and lots more...

Combinatorics: e.g. R.P. Stanley's Combinatorics and Commutative Algebra or his intro also in Lattices

If you want to get a feel for these and more, you could no better IMO than to pick up Weibel's History of Homological Algebra. If you want to start to learn this material I'd also suggest Weibel's Introduction to Homological Algebra.

Although I've never read it myself, this might interest you.

Schaum's Outlines - Linear Algebra provides a lot of useful proofs and theory behind abstract vector spaces if that's the kind of stuff you need. It also goes into Hermition forms and various complex applications

>Peter Lax turned out to be too concise for me.

You prefer a math text that rambled on and doesn't get to the point? I would think concision is a good thing in a text.

• Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach
  
  
• Div, Grad, Curl, and All That: An Informal Text on Vector Calculus
  
  
• Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds
  
  
• Advanced Calculus of Several Variables (Dover Books on Mathematics)
  
  
• Calculus of Several Variables (Undergraduate Texts in Mathematics)

Late, but here are undergrad books on the subject: geometric algebra, geometric calculus.

A grad-type book that has both and their applications to physics would be this one

I'm currently learning the geometric algebra undergrad book. It's a good read so far, and the author keeps up with book errors.

https://www.amazon.com/Vector-Calculus-Linear-Algebra-Differential/dp/0136574467

Plus you get to learn differential forms!

Linear Algebra can be of different levels of difficulty:

1. First encounter(proof based).
   
2. More advanced..
   
3. This will put hair on your chest..

Grad, curl, and div are essentially all the same operation: the exterior derivative.

Grad takes a scalarfield and gives you a vectorfield.

Curl takes a vectorfield and gives you a vectorfield.

Div takes a vectorfield and gives you a scalarfield.

But there's more to it. If you look at the resulting fields from these operations and then perform a change of variables (in manifold speak, you view the same fields in a different chart), they don't look right.

It turns out there is another kind of object called a k-form. These give to every point in space an alternating k-linear form on the tangent space. In other words, you give it k tangent vectors and it will spit out a number. The adjective alternating means if any tangent vector is repeated, the number output is zero. And the number is multilinear (ie: it's linear in each input separately, so doubling the length of any tangent vector doubles the number it spits out).

The 0-forms of a manifold are just scalarfields. The 1-forms are covectorfields. (You give a tangent vector and it spits out a number). As geometric objects, 1-forms look exactly like vectorfields, but they act different under a change of coordinates. You might say they transform "correctly".

A 1-form can be integrated over a curve. The result is a line integral, just like in usual vector calculus. However, because our objects now transform correctly, changing coordinates works as it should.

So grad, rather than being a mapping from scalarfields to vectorfields is actually a mapping from 0-forms (also just scalarfields) to 1-forms (covectorfields).

Similarly, curl isn't a mapping from vectorfields to vetorfields. Instead, it is a mapping from 1-forms to 2-forms.

A 2-form takes two tangent vectors and spits out a number. Intuitively, it returns the signed area spanned by the two tangent vectors.

Note that even though both 1-forms and 2-forms naively sync up with the notion of "vectorfield", they both act differently than a vectorfield and from each other. Under the hood, it has to do with the properties of the wedge product, written ∧, the basic operation for combining forms.

In R^3, we have standard basis x, y, and z. Well, it turns out that the k-forms also have a standard basis. For 1-forms, our basis is dx, dy, and dz. (The covector dx eats a vector and tells you what its x-component was, etc). For 2-forms, we just wedge things together: dx ∧ dy, dy ∧ dz, and dz ∧ dx form a basis. (One of our properties for wedge products is that a ∧ b = -(b ∧ a), so dy ∧ dx wouldn't be included in the basis if dx ∧ dy was). Meanwhile, 3-forms have a basis consisting of just of the triple-wedge dx ∧ dy ∧ dz.

Count the dimension of these spaces. At any point in our manifold, the 1-forms form a 3-dimensional vectorspace and the 2-forms also form a 3-dimenisonal vectorspace... but they have different bases. The 3-forms are 1-dimensional and (trivially) the 0-forms are also 1-dimensional. But again, they are not the same space!

So finally, div is a mapping from 2-forms to 3-forms.

The k-forms are admittedly very convoluted and tricky to work with. They are hardly intuitive compared to what you learn in vector calculus. But they have the advantage of playing nicely with change of coordinates. Maybe more importantly, they also generalize to any number of dimensions. Grad, curl, and div only really work in R^3. But the theory of electromagnetism and the theory of relativity take place in R^4.

For a nice introduction to the subject, you might want to check out Hubbard and Hubbard's excellent and pragmatic introduction to the subject.