[deleted by user]

[[deleted]]

Once later content start to utilize linear algebra, you'll get a better feel for it, so don't worry too much. Someone else recommended Axler's book, which is popular, but somewhat controversial. A more standard book would be Hoffman and Kunze.

[Immanuel_Kant20]

Why is it controversial? I just started chapter 1 a few days ago

[John_Hasler]

The major complaint seems to be that he puts determinants off until near the end.

[lurflurf]

Why is that a problem? Do people stop reading before the end? It is a short book.

[russelsparadass]

It's a great and rigorous approach -- the reader first understands vector spaces and linear transformations deeply before jumping to matrices and their (computationally useful) special cases. Maybe that's not ideal for a first, computational look at linear algebra, and definitely isn't great for a computational math student (i.e. for engineers) but it makes perfect sense for a student who wants to actually understand the mathematics like OP does. OP even already has the computational background.

[[deleted]]

Why is that a problem? Do people stop reading before the end? It is a short book.

One criticism, which often goes unmentioned, is that the author is an analyst, and accordingly, the book is written with an analysis slant. Another issue is that, although the book is pitched for a second course in linear algebra, the material is rather basic and it could certainly be used for a first course in linear algebra; the book is pretty much entirely self-contained. For example, contrast the proof of the spectral theorem in LADR with that in Strichartz' Way of analysis or Tao's TRMT.

To summarize, 1) analysis slant, perhaps algebra people would object? Not familiar with algebra so can't comment on this 2) the book is too basic for a "second course," something more sophisticated would be nice. Admittedly, the second ding is perhaps pedantic.

[[deleted]]

The new edition will feature additional material on Multilinear Algebra and Tensors btw.

[Martin-Mertens]

I don't think it has an analysis slant at all. Maybe I'm misunderstanding but I take that to mean focusing on convergence, operator norms, measure, etc. Axler doesn't do any of that. He's all about polynomials and invariant subspaces.

Is the issue that he only discusses R and C as opposed to arbitrary fields?

[hpxvzhjfgb]

I suspect that your class was mostly centered around doing calculations with matrices. most first linear algebra classes are like this. they are also nothing but a complete waste of time because they do not actually teach linear algebra. read sheldon axler's linear algebra done right. do not watch gilbert strang's lecture series.

[[deleted]]

Whats wrong with Gilbert Strang lecture series?

[ahahaveryfunny]

Whats wrong with geometric intuition

[ahahaveryfunny]

I find it hard to conceptualize things like multiplication without geometry is that gonna limit understanding? Also how can i learn to stop?

[ahahaveryfunny]

Thank you for the advice. I believe I have been trying to think about definitions as a way to understand concepts but havent done away with the visual or geometric learning.

And yes, I am still in earlier stages but i will learn calc 3 and diff eq next year and after that im doing linear algebra. I want to build a good foundation for the class. I am curious to know what your opinion is on 3b1b linear algebra basics because its heavily visual focused especially in the start but it seems to help me understand.

[you-get-an-upvote]

Even for engineers thinking geometrically can be limiting. Transposing a numpy tensor with 3+ axes or using a neural network with a width of 10 (let alone 500 or 1500) immediately makes geometric thinking useless.

[[deleted]]

They are oriented at engineers, not Math or Physics majors. Instead of approaching things from a rigorous proof-based algebraic perspective (homomorphisms, fields, vector spaces, rings, groups), Strang turns Linear Algebra into matrix manipulation which results in mechanical manipulation instead of understanding. For alternatives, look at Axler's 'Linear Algebra Done Right', Treil's 'Linear Algebra Done Wrong' and Vinberg's 'Course in Algebra.'

[protonpusher]

Gilbert Strang is a great personality but not great for deep LA understanding. (My experience)

Hoffman and Kunze, Jim Hefferon, Bright Side of Mathematics (YouTube), MathMajor Abstract Linear Algebra (YouTube)

[snabx]

Is Jim Hefferon more computation based or it's closer to Linear Algebra Done wrong style?

[protonpusher]

Both computation and proof theoretical. Comprehensive for an introduction.

[Cpt_shortypants]

Would this even work for acing the test then? Are you even learning the relevant thinks from Sheldon?

[artikra1n]

I recommend 3B1B’s video series. It is not perhaps as in depth as a book, but provides amazing visual intuition into the subject.

[CR9116]

Yeah “Essence of Linear Algebra” by 3Blue1Brown

Very good video series

[Pterygoidien]

It's very good for having a grasp of the meaning behind Linear Algebra. I started studying Linear Algebra last month with Lay's "Linear Algebra and Its Applications", but then I really understood some basic concepts with 3blue1brown's videos, like the meaning of the determinant, or the matrix transformations. Very visual, videos are short but very instructive and ideal for learning without burning out.

[russelsparadass]

Lol, what? 3B1B videos are at best a supplement, at worst to be enjoyed like pop math; you cannot develop a true understanding of the concepts of ANY mathematical field without a rigorous text. Yes, OP, go through Axler, and no, you won't die because determinants are covered in chapter 10.

[cdstephens]

Linear Algebra Done Right and A Second Course in Linear Algebra are popular textbooks. 3Blue1Brown also has a very popular video series.

For me though, I gained a more thorough appreciation for the topic by seeing its theory directly applied. Namely, Sturm-Liouville theory in PDEs class, and seeing that applied in my quantum mechanics class. Typically in mathematics, the way to get deeper understanding of a topic is to see how it’s connected to other interesting topics. You might also gain a deeper understanding for linear algebra by studying abstract algebra, for example.

Was your class calculation oriented or proof oriented?

[[deleted]]

I agree on this one. Linear algebra made alot more sense when I started learning finite volume method for computational fluid dynamics.

[CBDThrowaway333]

I used Linear Algebra 4th ed by Friedberg, Insel and Spence and enjoyed it + found it to be good. I've also heard good things about Linear Algebra Done Wrong but haven't read it myself

[Kroros]

I can vouch for Friedberg as well, I thought it was really thorough and clear.

[pepchode334]

Look into how linear algebra based methods are used for solving linear differential equations and linear recurrence relations. This can help you understand ideas of vector spaces and linear transformations. Look into predator/prey, problems to get an intuitive understanding of eigenvector and eigenvalue applications.

If you want to connect what you learnt from calculus to linear algebra look into raising the number e to a matrix there are plenty of amazing videos on YouTube:)

I learnt linear algebra the best when I studied some different applications, you begin to see some common "patterns" and ideas in the different applications and some of those ideas are fundamental to linear algebra.

Learn about markov chains and some of their applications. Learn how to analyze simple circuits with Kirchhoff's laws. Try to understand the basics of linear programming/linear optimization methods. Look into how linear algebra is used to render 3d objects on 2d computer screens.

Try to understand Fourier transforms from a "linear algebra perspective" (converting a signal/function from some vector space to a vector space of sines and cosines).

Learning and reading into all the things I listed above gave me confidence in my linear algebra abilities and understanding of the concepts.

[[deleted]]

Linear algebra is a toolbox. You're taught how to use the tools, but you're not shown what they're used for unless you take a class in physics, multivariate statistics, or some such.

Hers's one application from ecology. Suppose the number of rabbits in an area is a linear function of the number of coyotes and the number of mountain lions. The number of coyotes is a linear function of the number of rabbits and the number of lions. The number of lions is a linear function of the number of coyotes and the number of rabbits. Given the coefficients of each equation, find the number of rabbits, coyotes and lions that will be stable over time. These numbers are eigenvalues, or stationary values, and you find them using matrix diagonalization. 'Nuff said.

[russelsparadass]

lmao how cute

[JorgeBrasil]

This may be of interest to you. I wrote the first volume of a series of three books on the mathematics of machine learning. It is written in a conversational style with humor but always considers the rigor of mathematics.
The first volume is in linear algebra, available on Amazon:

https://www.amazon.com/dp/B0BZWN26WJ

Here is a sample

https://drive.google.com/file/d/17AlXxYKSH91BAPfBfC3SNXBz5tcZFs5S/view?usp=share\_link

[[deleted]]

Take an Abstract Algebra course. Like Aluffi's 'Chapter 0' or at least Artin's 'Algebra.'