Linear algebra
For courses in Advanced Linear Algebra. This top-selling, theorem-proof text presents a careful treatment of the principal topics of linear algebra, and illustrates the power of the subject through a variety of applications. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate.
Stats for nerds
10recommend
20mentions
17people
2008–2026years
60% recommend40% neutral
Mentions per year
What Hacker News says
- I found Linear Algebra by Friedberg, Insel and Spence to be excellent. Very clear, modern notation, great exercises. It's also what Tao lectured from in 115A: https://www.math.ucla.edu/~tao/resource/general/115a.3.02f/
- Linear Algebra and Geometry by Igor Shafarevich, coupled with Linear Algebra by Friedberg, Insel, and Spence; the latter has great problems to work with, whereas the former is for a lucid exposition.
- Gilbert Strang's "Intro to Linear Algebra"[1] is widely recommended and I enjoyed it as a supplement to Friedberg, Insel, and Spence's much more formal "Linear Algebra"[2].
[1] https://math.mit.edu/~gs/linearalgebra/ila6/indexila6.html
[2] https://www.pearson.com/en-us/subject-catalog/p/linear-algeb...
- Tao's notes seem to be based on the book Linear Algebra by Friedberg, Insel and Spence. I found it to be one of the best books on Linear Algebra, better than even Hoffman/Kunze. The proofs are extremely clear, it has examples like PageRank, Markov Chains, PCA and the solutions to just about every exercise is available on Quizlet.
- Supposing you have at least high school mathematics under the belt:
1. Book of Proof by Hammack [1]
2. Tom Apostol Calculus Vol 1 (stop at Linear Algebra)
3. Linear Algebra by Insel, Spence & Friedberg
4. Understanding Analysis by Stephen Abbott
5. Tom Apostol Calculus Vol 2
These are books with excellent exposition, and solution manuals are available when needed. You are absolutely required to do all the exercises. To learn mathematics is to do mathematics.
Two other excellent books are Spivak - Calculus, the problems in this book are very good, but much more difficult than Apostol. For more advanced analysis there's Walter Rudin's Principles of Mathematical Analysis (baby Rudin), which on its own is too difficult as a first exposure to analysis, but there's a cool professor that recorded an entire semester of baby Rudin lecture videos [2]. This is as close as it gets without being enrolled in a university.
[1] https://www.people.vcu.edu/~rhammack/BookOfProof/
[2] https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTk... (the first video is a bit blurry, it gets better)
- The best route is a community college. You can get most of this through a CC. Some offer discrete math, not many offer graph theory, and you probably won't get much astro (but you can get astronomy) or string theory from them. But it is good to get academic contacts who can give you direction. Someone who personally understands your drive and work ethic will have a better ability to give you suggestions. So this is my number one suggestion.
If you want to self learn, well let's go through some books and then youtube channels.
Books: Missing Discrete and Graph Theory
(You can get previous versions to save money. The content of these is mostly the same). Mostly in order of level (math then physics)
Calculus: Stewart's Calculus[1] (this is pretty much the standard) This has calc 1,2, and 3 (multi variable)
Linear Algebra: David Lay [2]. Start sometime after calc 2 (series problems). This will start you on some optimization and constraint solving. Stress learning eigen values/vectors and least squares. I don't have a good level 2 book, but that would mean looking into coordinate transformations, QR decomposition, and some more stuff.
Differential Equations: Blanchard Differential equations [3]. You will need diff eq to gain a true appreciation for physics. You will also gain a lot of the pre-req's for optimization and constraint solving.
Physics: Halliday and Resnik[4] is one of my favorites. But this is the lower college level (3 courses: Classical, E&M, Rel/quant). If you are relearning you can skip to below (though you might struggle a little more) Req: Taken or taking Calc 1 (differentiation and integration required later)
Classical Dynamics: Thornton[5] You will learn A LOT about constraint, optimization, and simple harmonic motion (necessary!!). You will also learn about Hamiltonian Systems. (1.5 courses) Req: Diff Eq, Calc 3
Electrodynamics: Griffiths [6]. Another standard. You won't find a better book than this for E&M. (1.5 courses) Req: Diff Eq, Calc 3
Quantum Mechanics: Griffiths [7] (He's the man, seriously) (1.5 courses) Req: Diff Eq, Calc 3 (lin algebra is nice, same with a tad of group theory)
Astrophysics: BOB [8] Lovingly called the "Big Orange Book" you will see this on every astrophysicists' shelves. (2+ courses) Req: Calc 1
Particle Physics: That's right! You guessed it! Griffiths![9] Take after QM.
Youtube:
BlackPenRedPen[10]: Fantastic teacher. He will help you with calc and help you understand a lot of tricks that you might not see in the above books. I can't stress enough that you should watch him.
Go find MIT OCWs, I'm not going to list them.
Honorable mentions: 3Blue1Brown[11], Numberphile[12], Veritasium[13], StandUpMaths[14], SmarterEveryDay[15]. All these people talk about some neat concepts that will help you gain more interest and think about things to pursue. But they are not course channels, they are much more casual (somewhere between what you'd see on the Discovery Channel and a classroom, more towards the latter).
Addendum:
> I follow a bunch of folks on the internet and idolize them for their multifaceted personalities
Don't stress too much about being like those people you idolize. I guarantee that you see them as much more intelligent people than they are or think of (not dissing on them, but we tend to put these people on pedestals and this is a big contributor to Imposter Syndrome. Which WILL have, probably already does, an effect on your learning process). Don't compare yourself. You can get to most of these peoples' levels by just doing an hour or two a day for a few years.
> I can totally see that these are the folks who have high IQs and they can easily learn a new domain in a few months if they were put in one.
This is a skill. A trainable skill. Just remember that. Some people are much more proficient at it, but I be you'll see that they have much more experience. In music you sight read. Doing the same thing with math, physics, engineering, etc will result in the same increase in talent.
[1] https://smile.amazon.com/Calculus-Early-Transcendentals-Jame...
[2] https://smile.amazon.com/Linear-Algebra-Its-Applications-3rd...
[3] https://smile.amazon.com/Differential-Equations-Tools-Printe...
[4] https://smile.amazon.com/Fundamentals-Physics-David-Halliday...
[5] https://smile.amazon.com/Classical-Dynamics-Particles-System...
[6] https://smile.amazon.com/Introduction-Electrodynamics-David-...
[7] https://smile.amazon.com/Introduction-Quantum-Mechanics-Davi...
[8] https://smile.amazon.com/Introduction-Modern-Astrophysics-Br...
[9] https://smile.amazon.com/Introduction-Elementary-Particles-D...
[10] https://www.youtube.com/channel/UC_SvYP0k05UKiJ_2ndB02IA
[11] https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
[12]https://www.youtube.com/channel/UCoxcjq-8xIDTYp3uz647V5A
[13] https://www.youtube.com/channel/UCHnyfMqiRRG1u-2MsSQLbXA
[14] https://www.youtube.com/user/standupmaths
[15] https://www.youtube.com/channel/UC6107grRI4m0o2-emgoDnAA
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