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# Reddit reviews on Linear Algebra Done Right (Undergraduate Texts in Mathematics)

Sentiment score: 21

Reddit mentions: 32

We found 32 Reddit mentions of Linear Algebra Done Right (Undergraduate Texts in Mathematics). Here are the top ones.

## Found 32 comments on Linear Algebra Done Right (Undergraduate Texts in Mathematics):

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-SolvingCalculusLinear AlgebraDifferential EquationsNumber TheoryProof-WritingAnalysisComplex AnalysisFunctional AnalysisPartial Differential EquationsHigher-dimensional Calculus and Differential GeometryAbstract AlgebraGeometryTopologySet Theory and LogicCombinatorics / Discrete MathGraph TheoryP. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"I doubt that you're going to find everything you're looking for in a single book.

I suggest that you start with Axler's Linear Algegra Done Right. Despite the pretentious name it does a good job of introducing linear algebra in a general form.

But Axler doesn't do any applications and almost completely ignores determinants (which I like, but it sounds like you want more of that) so I would supplement with Strang's MIT Lectures and any one of his books.

I would guess that career prospects are a little worse than CS for undergrad degrees, but since my main concern is where a phd in math will take me, you should get a second opinion on that.

Something to keep in mind is that "higher" math (the kind most students start to see around junior level) is in many ways very different from the stuff before. I hated calculus and doing calculations in general, and was pursuing a math minor because I thought it might help with job prospects, but when I got to the more abstract stuff, I loved it. It's easily possible that you'll enjoy both, I'm just pointing out that enjoying one doesn't necessarily imply enjoying the other. It's also worth noting that making the transition is not easy for most of us, and that if you struggle a lot when you first have to focus a lot of time on proving things, it shouldn't be taken as a signal to give up if you enjoy the material.

This wouldn't be necessary, but if you like, here are some books on abstract math topics that are aimed towards beginners you could look into to get a basic idea of what more abstract math is like:

Different mathematicians gravitate towards different subjects, so it's not easy to predict which you would enjoy more. I'm recommending these five because they were personally helpful to me a few years ago and I've read them in full, not because I don't think anyone can suggest better. And of course, you could just jump right into coursework like how most of us start. Best of luck!

(edit: can't count and thought five was four)

First, Sheldon Axler's

Linear Algebra Done Rightis a favorite of many folks. It might be a little much as supplementary material, but it will likely give you a very fresh perspective on whatever material you're covering in class.Second, disabuse yourself of the idea that "vectors" are little arrows sitting in some n-dimensional space. Vectors are not the important concept,

vector spacesare. A vector is just a name for an element of a vector space and they may or may not be representable as a little arrow or a finite n-tuple (x, x1, ..., x2). A vector space over the real numbers isnanycollection of objects which satisfy a particular set of properties.For example, the space of all continuous functions on the real line f: ℝ → ℝ, defined as

> C^(0)(ℝ) = {f: ℝ → ℝ | f is continuous}

is a vector space that doesn't look

anythinglike ℝ^(n) (n-dimensional Euclidean space). We can talk about maps between vector spaces which preserve the vector space structure, which we call linear maps or linear transformations. Matrices are one way of representing linear maps and matrix "multiplication" is defined so that the product of two matrices corresponds to the composition of the linear maps they represent.That fact is probably the first "ah ha!" moment for a lot of students that makes them realize there's more going on in linear algebra than just a bunch of vector/matrix manipulation.

In your first linear algebra course you'll likely be focusing on ℝ^(n), which has a bunch of extra structure relative to a plain vector space. It's very easy for a first-time linear algebra student to conflate all the extra "stuff" that comes with the structure of ℝ^(n) with the totality of linear algebra. So keep an eye out!

Third, you might be tempted to relate what you've done with matrices in the context of game programming to what you're learning in your linear algebra class as quickly as possible. Be careful because the way computers use matrices is slightly more complex than the material you'll learn early on. Specifically, folks tend to use homogeneous coordinates when writing games, which is why libraries like OpenGL use 4×4 matrices to represent transformations of 3d space instead of 3×3 matrices (and, similarly, 3×3 matrices to represent transformations of 2d space instead of 2×2 matrices as you might naïvely expect).

We do this because computers (GPUs, really) are built to manipulate matrices very, very quickly, so we ♥ it when the transforms we want to perform can be express as a matrix. Unfortunately, translations in n-dimensional space can't be represented as n×n matrices. However, if we move to real projective space then there is a way to represent translations as matrices in the projective space. But points in n-dimensional projective space can be represented by n+1 coordinates, so while there is no n×n matrix representing a translations in n-dimensional space there are corresponding (n+1)×(n+1) matrices which represent that translation in n-dimensional projective space.

There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.

To illustrate my point:

Linear Algebra:

Linear Algebra Through Geometry by Banchoff and Wermer

3. Here's more rigorous/abstract Linear Algebra for undergrads:

Linear Algebra Done Right by Axler

4. Here's more advanced grad level Linear Algebra:

Advanced Linear Algebra by Steven Roman

-----------------------------------------------------------

Calculus:

Calulus by Spivak

3. Full-blown undergrad level Analysis(proof-based):

Analysis by Rudin

4. More advanced Calculus for advance undergrads and grad students:

Advanced Calculus by Sternberg and Loomis

The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning

mathright now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.Here's how you start studying real math NOW:

Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into

Discrete Math by Susanna Epp

How To prove It by Velleman

Intro To Category Theory by Lawvere and Schnauel

There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.

If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:

Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.

I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.

Good Luck, buddyroo.

You're not really doing higher math right now as much as you're learning tricks to solve problems. Once you start proving stuff that'll be a big jump. Usually people start doing that around Real Analysis like your father said. Higher math classes almost entirely consist of proofs. It's a lot of fun once you get the hang of it, but if you've never done it much before it can be jarring to learn how. The goal is to develop mathematical maturity.

Start learning some geometry proofs or pick up a book called "Calculus" by Spivak if you want to start proving stuff now. The Spivak book will give you a massive head start if you read it before college. Differential equations will feel like a joke after this book. It's called calculus but it's really more like real analysis for beginners with a lot of the harder stuff cut out. If you can get through the first 8 chapters or so, which are the hardest ones, you'll understand a lot of mathematics much more deeply than you do now. I'd also look into a book called Linear Algebra done right. This one might be harder to jump into at first but it's overall easier than the other book.

>When university starts, what can I do to ensure that I can compete and am just as good as the best mathematics students?

Read textbooks for mathematics students.

For example for Linear Algebra I heard that Axler's book is very good (I studied Linear Algebra in another language, so I can't really suggest anything from personal experience). For Calculus I personally suggest Spivak's book.

There are many books that I could suggest, but one of the greatest books I've ever read is The Art and Craft of Problem Solving.

Hrrumph. Determinants are a capstone, not a cornerstone, of Linear Algebra.

https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582

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&amp;ie=UTF8&amp;qid=1413311074&amp;sr=1-1&amp;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.

Linear algebra is an essential tool in many areas of mathematics. Computations with matrices aren't always that important; far more important are the concepts of vector space and linear transformation. Pretty much any time you work with coordinates, dimension, changes of coordinates, vectors, linear relations, or anything like that, you're going to need some linear algebra.

If you're interested, I recommend taking a look at Axler's

Linear Algebra Done Right. Axler has very clear exposition and proofs, and if you've only seen the computational aspect of linear algebra, it'll provide a different, more abstract and conceptual perspective.Hey! I am a math major at Harvey Mudd College (who went to high school in the Pacific NW!). I'll answer from what I've seen.

End: Also, if you wanna learn something cool, I'd check out Discrete math. It's usually required for both a math or CS major, and it's some of the coolest undergraduate math out there. Oh, and, unlike some other math, it's not terrible to self-teach. :)

Good luck! Math is awesome!

This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.

General Stuff:The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.

Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.

Axler's Linear Algebra Done Right is something you might enjoy looking at; since his basic point of view is that linear algebra is generally done wrong.

http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582

A First Course in Graph Theory by Chartrand and Zhang

Combinatorics: A Guided Tour by Mazur

Discrete Math by Epp

For Linear Algebra I like these below:

Lecture Notes by Tao

Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza

Linear Algebra Done Right by Axler

Linear Algebra by Friedberg, Insel and Spence

You have a long way to go. Here's a path that will get you to (smooth) manifold theory, with book suggestions:

I'd suggest Probability, Linear Algebra, Convex Optimization and ML in that order.

As for study materials, I'd suggest

That should keep you busy for a while.

I learned lin. alg. from Axler's

Linear Algebra Done Right. I found it extremely readable, with exercises that were not too hard to get through quickly.There are some really good books that you can use to give yourself a solid foundation for further self-study in mathematics. I've used them myself. The great thing about this type of book is that you can just do the exercises from one side of the book to the other and then be confident in the knowledge that you understand the material. It's nice! Here are my recommendations:

First off, three books on the basics of algebra, trigonometry, and functions and graphs. They're all by a guy called Israel Gelfand, and they're good: Algebra, Trigonometry, and Functions and Graphs.

Next, one of two books (they occupy the same niche, material-wise) on general proof and problem-solving methods. These get you in the headspace of constructing proofs, which is really good. As someone with a bachelors in math, it's disheartening to see that proofs are misunderstood and often disliked by students. The whole point of learning and understanding proofs (and reproducing them yourself) is so that you gain an understanding of the

whyof the problem under consideration, not just thehow... Anyways, I'm rambling! Here they are: How To Prove It: A Structured Approach and How To Solve It.And finally a book which is a little bit more terse than the others, but which serves to reinforce the key concepts: Basic Mathematics.

After that you have the basics needed to take on any math textbook you like really - beginning from the foundational subjects and working your way upwards, of course. For example, if you wanted to improve your linear algebra skills (e.g. suppose you wanted to learn a bit of machine learning) you could just study a textbook like Linear Algebra Done Right.

The hard part about this method is that it takes a lot of practice to get used to learning from a book. But that's also the upside of it because whenever you're studying it, you're

reallystudying it. It's a pretty straightforward process (bar the moments of frustration, of course).If you have any other questions about learning math, shoot me a PM. :)

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.

If you're dislike of linear algebra comes from using the determinant and matrix calculations, you would love Axler's Linear Algebra Done Right.

Some possibilities:

Calc I & II: Spivak's

CalculusCalc III and a bit of linear algebra: Hubbard & Hubbard's

Vector CalculusLA: Axler or Shilov or both

ODE: Morris Tanenbaum

Discrete/Combinatorics/etc.: Knuth's

Concrete MathematicsFor book suggestions beyond concerning Analysis, Algebra, and Topology, the search box will turn up a ton of previous conversations.

It's aight. Just read linear algebra, and mv calculus. Maybe some statistical mechanics, read some thermo and kinetics. Atkins for kinetics and thermo, McQuarrie for stat mech. For linear algebra read get this. You'll still have to take classes on it, so it's cool. The worst you may have to do is take some UG classes to get up to speed.

Linear algebra is about is about linear functions and is typically taken in the first or second year of college. College algebra normally refers to a remedial class that covers what most people do in high school. I highly recommend watching this series of videos for getting an intuitive idea of linear algebra no matter what book you go with. The book you should use depends on how comfortable you are with proofs and what your goal is. If you just want to know how to calculate and apply it, I've heard Strang's book with the accompanying MIT opencourseware course is good. This book also looks good if you're mostly interested in programming applications. A more abstract book like Linear Algebra Done Right or Linear Algebra Done Wrong would probably be more useful if you were familiar with mathematical proofs beforehand. How to Prove it is a good choice for learning this.

I haven't seen boolean algebra used to refer to an entire course, but if you want to learn logic and some proof techniques you could look at How to Prove it.

Most calculus books cover both differential and integral calculus. Differential calculus refers to taking derivatives. A derivative essentially tells you how rapidly a function changes at a certain point. Integral calculus covers finding areas under curves(aka definite integrals) and their relationship with derivatives. This series gives some excellent explanations for most of the ideas in calculus.

Analysis is more advanced, and is typically only done by math majors. You can think of it as calculus with complete proofs for everything and more abstraction. I would not recommend trying to learn this without having a strong understanding of calculus first. Spivak's Calculus is a good compromise between full on analysis and a standard calculus class. It's possible to use this as a first exposure to calculus, but it would be difficult.

I've never taught the course, but a couple of my colleagues are very fond of Linear Algebra Done Wrong and would willingly teach from it if (1) the title wouldn't immediately turn students off of it and (2) the school would be okay with sacrificing some income from students having to purchase a book.

If you're curious, the book title is a play on the title of another well-known linear algebra book.

You could consider starting with a book like Velleman's How to Prove It. It doesn't have to be that book, there are also free options online, but learning some logic and set theory from a book like that is a good way to figure out how to work with the other subjects you're working on.

Then, you could find a rigorous treatment of the subjects you want to learn. Something like Axler's Linear Algebra Done Right or Spivak's Calculus.

Learning math from textbooks like this is harder, but you end up with a better understanding of the math.

You can pick up an intro book on Linear Algebra (eg: Linear Algebra Done Right) and read about "Inner Product Spaces". In finite dimensions, these are essentially the same thing as Hilbert Spaces, whereas Banach Spaces would just be "Normed Spaces". Inner Product Spaces have a lot more extra tools that you can use that are really powerful, compared to normed spaces.

You can also just read about Lp-Spaces on Wikipedia. All Lp-spaces are Banach Spaces, by only L2 spaces are Hilbert Spaces, and there you get things like Fourier Transforms, which you don't have in general.

Linear algebra is about vector spaces and linear functions, which are a nice kind of functions. It has many applications in pure and applied math. Some people say you can never know enough linear algebra, and I'm inclined to agree. Abstract algebra is, well, abstract. You should probably do some reading for yourself. If you'd like book recommendations, Charles C. Pinter's A Book of Abstract Algebra is most excellent, as is Linear Algebra Done Right by Axler.

Linear Algebra can be of different levels of difficulty:

Honestly, I think you should be more realistic: doing everything in that imgur link would be insane.

You should try to get a survey of the first 3 semesters of calculus, learn a bit of linear algebra perhaps from this book, and learn about reading and writing proofs with a book like this. If you still have time, Munkres'

Topology, Dummit and Foote'sAbstract Algebra, and/or Rudin'sPrinciples of Mathematical Analysiswould be good places to go.Roughly speaking, you can theoretically do intro to proofs and linear algebra independently of calculus, and you only need intro to proofs to go into topology (though calculus and analysis would be desirable), and you only need linear algebra and intro to proofs to go into abstract algebra. For analysis, you need both calculus and intro to proofs.

I think linear algebra is a much more interesting topic without getting bogged down in matrix computations, such as what Axler does with Linear Algebra Done Right. That's just my opinion I suppose.

As others mentioned, it is very hard to make progress learning programming without using a computer (think of reading about driving without ever driving a car). Instead, get yourself excited about science and computer science:

Science:

Computer Science (actually math, but this will help change the way you think to be more analytical, and will be useful for programming, vector graphics, etc.):

Don't skip proofs and wrestle through them. That's the only way; to struggle. Learning mathematics is generally a bit of a fight.

It's also true that computation theory is essentially all proofs. (Specifically, constructive proofs by contradiction).

You could try a book like this: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108/ref=sr_1_1?ie=UTF8&amp;qid=1537570440&amp;sr=8-1&amp;keywords=book+of+proof

But I think these books won't really make you proficient, just more familiar with the basics. To become proficient, you should write proofs in a proper rigorous setting for proper material.

Sheldon Axler's "Linear Algebra Done Right" is really what taught me to properly do a proof. Also, I'm sure you don't really understand Linear Algebra, as will become

veryapparent if you read his book. I believe it's also targeted towards students who have seen linear algebra in an applied setting, but never rigorous and are new to proof-writing. That is, it's meant just for people like you.The book will surely benefit you in time. Both in better understanding linear algebra and computer science

classicslike isomorphisms and in becoming proficient at reading/understanding a mathematical texts and writing proofs to show it.I strongly recommend the second addition over the third addition. You can also find a solutions PDF for it online. Try Library Genesis. You don't need to read the entire book, just the first half and you should be well-prepared.