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Yeah, a good example is on the second page of the first chapter: > Remark. It is easy to prove that zero vector 0 is unique, and that given v ∈ V its additive

Yeah, a good example is on the second page of the first chapter: > Remark. It is easy to prove that zero vector 0 is unique, and that given v ∈ V its additive inverse −v is also unique. The is the first time the word "unique" is used in the text. Students are going to have no idea whether this is meant in some technical sense or just conventional English. One can imagine various meanings, but that doesn't substitute for real understanding. This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. On the surface the opposite is true - you complain, for instance, that the text jumps immediately into using technical language without any prior introduction or intuition building. My take is that intuition building doesn't need to replace or preface the use of formal precision, but that what is needed is to bridge concepts the student already understands and has intuition for to the new concept that the student is to learn. In terms of intuition building, I think it's probably best to introduce vectors via talking about Euclidean space - which gives the student the possibility of using their physical intuitions. The student should build intuition for how and why vector space "axioms" hold by learning that fundamental operations like addition (which they already grasp) are being extended to vectors in Euclidean space. They already instinctively understand the axiomatic properties being introduced, it's just that the raw technical language being thrown at them fails to connect to any concept they already possess.

newprint 3y ago
> Remark. It is easy to prove that zero vector 0 is unique, and that given v ∈ V its additive inverse −v is also unique. I'm sorry, this book is meant for the audience who can read and write proofs. Uniqueness proofs are staple of mathematics. If word "unique" throws you off, then this book is not meant for you.
- curiousgal 3y ago
  No offense to OP but you are right. I get the feeling that people keep looking for a math-free resource to learn math...
  - CalChris 3y ago
    Now that would be unique.
    - chaboud 3y ago
      It'll be the publication of the uncompleteness theorem... representation of math with no math.
  - RoyalHenOil 3y ago
    I think a lot of people just need an opportunity to see math demonstrated in a more tangible way. For example, I learned trig, calculus, and statistics from my science classes, not from my math classes (and that's despite getting perfect A's in all of my math classes). In math class, I was just mindlessly going through the motions and hating every second of it, but science classes actually taught me why it worked and showed me the beauty and cleverness of it all. I just needed to see the math to grok it.
    - chongli 3y ago
      If you study mathematics at a rigorous level then you learn by writing proofs. Then you will rack your brain for hours or even days trying to figure out how to prove some simple things. It is not at all “going through the motions” at that point!
    - gosub100 3y ago
      I think most college math depts have "applied math" majors. I like both sides of math, but I found it incredibly frustrating when I would try to study just the equations for that chapter, only to be tested on a word problem. The whole "trying to trick you" conspiracy turned me off to college in general. If I'm trying to teach someone how to do something, I would show them "A, then, B, and you get C" , then assign a variety of homework of that form, and on the test, say "A, then B, then _____" and they would be correct if they concluded C. But for some reason this method isn't used much in university. If I wanted to teach a student how to start with C and deconstruct into A, B , thats what I would have taught them!
- wtallis 3y ago
  I'd go a bit further and say that if you're not comfortable with the basics of mathematical proofs, then you're not ready for the subject of linear algebra regardless of what book or course you're trying to learn from. The purely computational approach to mathematics used up through high school (with the oddball exception of Euclidean geometry) and many introductory calculus classes can't really go much further than that.
  - lll-o-lll 3y ago
    Or, you know, mathematics can be viewed as a powerful set of tools… Somehow I seem to remember getting through an engineering degree, taking all the optional extra math courses (including linear algebra), without there ever being a big emphasis on proofs. I’m sure it’s important if you want to be a mathematician, but if you just want to understand enough to be able to use it?
    - Koshkin 3y ago
      Sure, that's why we have "engineering mathematics" courses. This is not one of them.
      - lll-o-lll 3y ago
        Linear Algebra? I explicitly called it out there. Also, it wasn’t taught by the engineering faculty. I think your dismissive attitude is telling.
        dchftcs 3y ago
        I'm incredulous that a linear algebra course taught by mathematics faculty didn't have a lot of theorem proving. Maybe that would be the case if the intended audience is engineering students. But for mathematics students, it would literally be setting them up for failure; a student that can't handle or haven't seen much theorem-proving in linear algebra is not going to go very far in coursework elsewhere. Theorem proving is an integral part of mathematics, in stretching and expanding tools and concepts for your own use. Maybe the courses are structured so that mathematics students normally go on to take a different course. In that case, GP's point would still have been valid - the LA courses you took were indeed ones planned for engineering, not for those pursuing mathematics degrees. At my alma mater, it was indeed the case that physics students and engineering students were exposed to a different set of course material for foundational courses like linear algebra and complex analysis. Just like compiler theory, if you don't write compilers maybe it's not that useful and you shouldn't be spending too much time on it, but it would be presumptuous to say that delivering a full compiler course is a fundamentally incorrect approach, because somebody has to make that sausage.
- BOOSTERHIDROGEN 3y ago
  Any books how to read and write proofs ? The above statement of zero vector is unique, I have no idea what is that means.
  - navanchauhan 3y ago
    I would recommend Book of Proof: https://www.people.vcu.edu/~rhammack/BookOfProof/ https://www.people.vcu.edu/~rhammack/BookOfProof/ We used this in my Discrete Mathematics class (MATH 2001 @ CU Boulder) (it is a pre-requisite for most math classes). The section about truth tables did overlap a bit with my philosophy class (PHIL 1440 Critical Thinking)
  - mananaysiempre 3y ago
    > The above statement of zero vector is unique, I have no idea what is that means. In isolation, nothing. (Neither does the word “vector”, really.) In the context of that book, the idea is more or less as follows: Suppose you are playing a game. That game involves things called “vectors”, which are completely opaque to you. (I’m being serious here. If you’ve encountered about some other thing called “vectors”, forget about it—at least until you get to the examples section, where various ways to implement the game are discussed.) There’s a way to make a new vector given two existing ones (denoted + and called “addition”, but not the same as real-number addition) and a way to make a new vector given an existing one and a real number (denoted by juxtaposition and called “multiplication”, but once again that’s a pun whose usefulness will only become apparent later) (we won’t actually need that one here). The inner workings of these operations in turn are also completely opaque to you. However, the rules of the game tell you that 1. It doesn’t matter in which order you feed your two vectors into the “addition” operation (“add” them): whatever existing vectors v and w you’re holding, the new vector v+w will turn out to be the same as the other new vector w+v. 2. When you “add” two vectors and then “add” the third to the result, you’ll get the exact same thing as when you “add” the first to the “sum” of the second and third; that is, whatever the vectors u, v, and w are, (u+v)+w is equal to u+(v+w). (Why three vectors and not four or five? It turns out that you have the rule for three, you can prove those for four, five, and so on, even though there are going to be many more ways to place the parens there. See Spivak’s “Calculus” for a nice explanation, or if you like compilers, look up “reassociation”.) 3. There is [at least one] vector, call it 0, such that adding it to anything else doesn’t make a difference: for this distinguished 0 and whatever v, v+0 is the same as v. Let’s now pause for a moment and split the last item into two parts. We’ll say a vector u deserves to be called a “zero” if, whatever other vector we take [including u itself!], we will get it back again if we add u to it; that is, for any v we’ll get v+u=v. This is not an additional rule. It doesn’t actually tell us anything. It’s just a label we chose to use. We don’t even know if there are any of those “zeros” around! And now we can restate rule 3, which is a rule: 3. There is [at least one] “zero”. What the remark says is that, given these definitions and the three rules, you can show, without assuming anything else, that there is exactly one “zero”. (OK, what the remark actually says is that you can prove that from the full set of eight rules that the author gives. But that is, frankly, sloppy, because the way rule 4 is phrased actually assumes that the zero is unique: either you need to say that there’s a distinguished zero such that for every v there’s a w with v+w= that zero, or you need to say that for every v there’s a w such that v+w is a zero, possibly a different one for each v. Of course, it doesn’t actually matter!—there can only be one zero even before we get to rule 4. But not making note of that is, again, sloppy. This kind of sloppiness is perfectly acceptable among people who have seen this sort of thing before, say done finite groups or something like that. But if the book is supposed to be give a first impression, this seems like a bad idea. Perhaps a precalculus course of some sort is assumed. Read Spivak, seriously. He’s great. Not linear algebra, though.)
- margalabargala 3y ago
  That's fine, but then maybe this book shouldn't then be the top comment for being recommended as a "good first course" then?
  - Koshkin 3y ago
    Well, perhaps it's not meant to be the first book on mathematics one sees in their life.
    - echion 3y ago
      It's supposed to be the introduction to rigorous proof, so it is supposed to be "the first book on [formal] mathematics" for its audience.
    - margalabargala 3y ago
      From the second paragraph of the introduction to the book we are discussing: > Besides being a first course in linear algebra it is also supposed to be a first course introducing a student to rigorous proof, formal definitions---in short, to the style of modern theoretical (abstract) mathematics. So it's certainly meant to be the first math book one sees in their life that discusses rigorous proofs.
- echion 3y ago
  > this book is meant for the audience who can read and write proofs It seems like the opposite is true: "It is intended for a student who, while not yet very familiar with abstract reasoning, is willing to study more [than a] "cookbook style" calculus type course." (from the link). If your point is one can't learn linear algebra before learning "abstract [mathematical] reasoning"...don't think you're the main target audience of a subject as practical as linear algebra.
  - bscphil 3y ago
    It also goes on to say > Besides being a first course in linear algebra it is also supposed to be a first course introducing a student to rigorous proof, formal definitions---in short, to the style of modern theoretical (abstract) mathematics. So I think it's fair to say that the book (ought to) assume zero knowledge of proofs, contra your parent's claim that the audience is expected to be able to read and write proofs.
- itsoktocry 3y ago
  "Here's a great first course in Linear Algebra!" "If you can't read and write mathematical proofs, this isn't for you." Math snobs are funny, and are fully to blame for people hating math.
  - Sharlin 3y ago
    A first course in linear algebra still assumes background information, because linear algebra is not a basic topic. It’s not meant to be a first course in math. Math builds on itself and it would be incredibly inconvenient if every course everywhere would have to include a recap of basic things. And proofs are among the most fundamental things in math! Programming courses or articles or books, beyond the 101 level, don’t teach you again and again the basics of declaring a variable and writing a loop either! No field does that. Wrt linear algebra in particular, there are plenty of resources aimed at programmers thanks to its relevance in computer graphics and so on. They typically skip proofs and just tell you that this is how matrix multiplication is defined, but they don’t teach you math, merely using math. Which can be plenty enough to an engineer.
  - eigenket 3y ago
    This is not snobbery, some subjects just have prerequisites. You can't learn computer science without having a good sense of what an "algorithm" is, you have to know how to read and write and understand algorithms. Similarly you can't learn math without having a good sense of what a proof is, reading, writing and understanding proofs is the heart of what math is. Even more strongly, trying to learn math without a solid understanding of how proofs work is something like studying English literature while refusing to learn how to read English.
    - margalabargala 3y ago
      > Even more strongly, trying to learn math without a solid understanding of how proofs work is something like studying English literature while refusing to learn how to read English. It depends why you're trying to learn math. Are you interested in math for math's sake, or are you trying to actually do something with it? If it's the former, then yeah, you need proofs. Otherwise, like in your analogy, it's like studying English literature without knowing any english grammar rules. But if you're trying to apply the math, if you're studying linear algebra because it's useful rather than for its own sake, then you don't need proofs. To follow the same analogy, it's like learning enough English to be conversational and get around America, without knowing what an "appositive" is. The software industry, similarly, is full of people who make use of computer science concepts, without having rigorously studying computer science. You can't learn true "computer science" without an understanding of discrete math, but you can certainly get a job as an entry-level SWE without one. You don't need discrete math to learn python, see that it's useful, and do something interesting with it. The same applies to linear algebra. Everyone who does vector math doesn't need to be able to prove that the tools they are using work. If everyone who does vector math is re-deriving their math from first principles, then something's gone terribly wrong. There's a menu of known, well-defined treatments that can be applied to vectors, and one can read about them and trust that they work without having proven why they work. EDIT: it occurs to me, an even stronger analogy of this point, is that it is entirely possible to study computer science, without having any understanding of electrical engineering or knowing how a transistor works.
BoiledCabbage 3y ago
> This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. The thing that mathematicians refuse to admit is that they are extremely sloppy with their notation, terminology and rigor. Especially in comparison to the average programmer. They are conceptually/abstractly rigorous, but in "implementation" are incredibly sloppy. But they've been in that world so long they can't really see it / just expect it. And if you debate with one long enough, they'll eventually concede and say something along the lines of "well math evolved being written on paper and conciseness was important so that took priority over those other concerns." And it leaks through into math instruction and general math text writing. Programming is forced to be extremely rigorous at the implementation level simply because what is written must be executed. Now engineering abstraction is extremely conceptually sloppy and if it works it's often deemed "good enough". And math generally is the exact opposite. Even for a simple case, take the number of symbols that have context sensitive meanings and mathematicians. They will use them without declaring which context they are using, and a reader is simply supposed to infer correctly. It's actually somewhat funny because it's not at all how they see themselves.
- rq1 3y ago
  This is ridiculous. The average computer scientist (not only "programmer", as a js dev would be) never wrote lean/coq or similar, and is not aware of the Curry-Haskell like theorems and their implications.
  - thrwayaistartup 3y ago
    I think you entirely missed the point. GP put it well: >> They are conceptually/abstractly rigorous, but in "implementation" are incredibly sloppy. Maturity in concept-space and the ability to reason abstractly can be achieved without the sort of formal rigor required by far less abstract and much more conceptually simple programming. I have seen this first hand TAing and tutoring CS1. I regularly had students who put off their required programming course until senior year. As a result, some were well into graduate-level mathematics and at the top of their class but struggled deeply with the rigor required in implementation. Think about, e.g., missing semi-colons at the end of lines, understanding where a variable is defined, understanding how nested loops work, simple recursion, and so on. Consider something as simple as writing a C/Java program that reads lines from a file, parses them according to a simple format, prints out some accumulated value from the process, and handles common errors appropriately. Programming requires a lot more formal rigor than mathematical proof writing.
ravi-delia 3y ago
Axler serves as an adequate first introduction to linear algebra (though it is intended to be a second, more formal, pass through. Think analysis vs calculus), but it isn't intended to be a first introduction to all of formal mathematics! A necessary prereq is understanding some formal language used in mathematics- what unique means is included in that. Falling entirely back on physical intuition is fine for students who will use linear algebra only in physical contexts, but linear algebra is often a stepping stone towards more general abstract algebra. That's what Axler aims to help with, and with arbitrary (for instance) rings there isn't a nice spacial metaphor to help you. There you need to have developed the skill of looking at a definition and parsing out what an object is from that.
Tazerenix 3y ago
Mathematicians are well aware of complaints like these about introductions to their subjects, by the way. It is for a reason that this book introduces the theory of abstract vector spaces and linear transformations, rather than relying on the crutch of intuition from Euclidean space. If you want to become a serious mathematician (and this is a book for such people, not for people looking for a gentle introduction to linear algebra for the purposes of applications) at some point it is necessary to rip the bandaid of unabstracted thinking off and engage seriously with abstraction as a tool. It is an important and powerful skill to be presented with an abstract definition, only loosely related to concrete structures you have seen before, and work with it. In mathematics this begins with linear algebra, and then with abstract algebra, real analysis and topology, and eventually more advanced subjects like differential geometry. It's difficult to explain to someone whose exposure to serious mathematics is mostly on the periphery that being exposed forcefully to this kind of thinking is a critical step to be able to make great leaps forward in the future. Brilliant developments of mathematics like, for example, the realisation that "space" is an intrinsic concept and geometry may be done without reference to an ambient Euclidean space begin with learning this kind of abstract thinking. It is easy to take for granted the fruits of this abstraction now, after the hard work has already been put in by others to develop it, and think that the best way to learn it is to return back to the concrete and avoid the abstract.
- fakecontent000 3y ago
  God I just hate you so much.
- resource0x 3y ago
  >... rather than relying on the crutch of intuition from Euclidean space Euclidean space is not a good crutch, but there are other, much more meaningful, crutches available, like (orthogonal) polynomials, Fourier series etc. Not mentioning any motivations/applications is a pedagogical mistake IMO. I think we need some platform for creating annotated versions of math books (as a community project) - that could really help.
  - Tazerenix 3y ago
Ridj48dhsnsh 3y ago
I absolutely agree about additional rigor and precision making math easier to learn. Only after you're familiar with the concepts can you be more lazy. That's the approach taken by my favorite math book: https://people.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf https://people.math.harvard.edu/~shlomo/docs/Advanced_Calcul...
Galanwe 3y ago
> This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. This is _precisely_ the opinion of Roger Godement, French mathematician and member of the Bourbaky group. I would highly recommend his books on Algebra. They are absolutely uncompromising on precision and correctness, while also being intuitive and laying down all the logical foundations of their rigor. Overall, I cannot recommend enough the books of the Bourbaky group (esp. Dieudonne & Godement). They are a work of art in the same sense that TAOCP is for computer science.
- ykonstant 3y ago
  Unfortunately, some of the Bourbaki books need to be read in French, because the typesetting on the English translations is so atrocious as to be unreadable; as a consolation, the typesetting on the original French is, as always, immaculate.
seanhunter 3y ago
> This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. On the surface the opposite is true - you complain, for instance, that the text jumps immediately into using technical language without any prior introduction or intuition building. My take is that intuition building doesn't need to replace or preface the use of formal precision, but that what is needed is to bridge concepts the student already understands and has intuition for to the new concept that the student is to learn. If you read the book in the original post you may find it's absolutely for you. Axler assumes you know only the real numbers, then starts by introducing the commutative and associative properties and the additive and multiplicative identity of the complex numbers[1]. Then he introduces fields and shows that, hey look we have already proved that the real and complex numbers are fields because we've established exactly the properties required. Then he goes on to multidimensional fields and proves the same properties (commutativity and associativity and identities) in F^n where F is any arbitrary field, so could be either the real or the complex numbers. Then he moves onto vectors and then onto linear maps. It's literally chapter 3 before you see [ ] notation or anything that looks like a matrix, and he introduces the concept of matrices formally in terms of the concepts he has built up piece by piece before. Axler really does a great job (imo) of this kind of bridge building, and it is absolutely rigorous each step of the way. As an example, he (famously) doesn't introduce determinants until the last chapter because he feels they are counterintuitive for most people and you need most of the foundation of linear algebra to understand them properly. So he builds up all of linear algebra fully rigorously without determinants first and then introduces them at the end. [1] eg he proves that there is only one zero and one "one" such that A = 1*A and A = 0 + A.

newprint 3y ago

> Remark. It is easy to prove that zero vector 0 is unique, and that given v ∈ V its additive inverse −v is also unique. I'm sorry, this book is meant for the audience who can read and write proofs. Uniqueness proofs are staple of mathematics. If word "unique" throws you off, then this book is not meant for you.

curiousgal 3y ago
No offense to OP but you are right. I get the feeling that people keep looking for a math-free resource to learn math...
- CalChris 3y ago
  Now that would be unique.
  - chaboud 3y ago
    It'll be the publication of the uncompleteness theorem... representation of math with no math.
- RoyalHenOil 3y ago
  I think a lot of people just need an opportunity to see math demonstrated in a more tangible way. For example, I learned trig, calculus, and statistics from my science classes, not from my math classes (and that's despite getting perfect A's in all of my math classes). In math class, I was just mindlessly going through the motions and hating every second of it, but science classes actually taught me why it worked and showed me the beauty and cleverness of it all. I just needed to see the math to grok it.
  - chongli 3y ago
    If you study mathematics at a rigorous level then you learn by writing proofs. Then you will rack your brain for hours or even days trying to figure out how to prove some simple things. It is not at all “going through the motions” at that point!
  - gosub100 3y ago
    I think most college math depts have "applied math" majors. I like both sides of math, but I found it incredibly frustrating when I would try to study just the equations for that chapter, only to be tested on a word problem. The whole "trying to trick you" conspiracy turned me off to college in general. If I'm trying to teach someone how to do something, I would show them "A, then, B, and you get C" , then assign a variety of homework of that form, and on the test, say "A, then B, then _____" and they would be correct if they concluded C. But for some reason this method isn't used much in university. If I wanted to teach a student how to start with C and deconstruct into A, B , thats what I would have taught them!
wtallis 3y ago
I'd go a bit further and say that if you're not comfortable with the basics of mathematical proofs, then you're not ready for the subject of linear algebra regardless of what book or course you're trying to learn from. The purely computational approach to mathematics used up through high school (with the oddball exception of Euclidean geometry) and many introductory calculus classes can't really go much further than that.
- lll-o-lll 3y ago
  Or, you know, mathematics can be viewed as a powerful set of tools… Somehow I seem to remember getting through an engineering degree, taking all the optional extra math courses (including linear algebra), without there ever being a big emphasis on proofs. I’m sure it’s important if you want to be a mathematician, but if you just want to understand enough to be able to use it?
  - Koshkin 3y ago
    Sure, that's why we have "engineering mathematics" courses. This is not one of them.
    - lll-o-lll 3y ago
      Linear Algebra? I explicitly called it out there. Also, it wasn’t taught by the engineering faculty. I think your dismissive attitude is telling.
      - dchftcs 3y ago
        I'm incredulous that a linear algebra course taught by mathematics faculty didn't have a lot of theorem proving. Maybe that would be the case if the intended audience is engineering students. But for mathematics students, it would literally be setting them up for failure; a student that can't handle or haven't seen much theorem-proving in linear algebra is not going to go very far in coursework elsewhere. Theorem proving is an integral part of mathematics, in stretching and expanding tools and concepts for your own use. Maybe the courses are structured so that mathematics students normally go on to take a different course. In that case, GP's point would still have been valid - the LA courses you took were indeed ones planned for engineering, not for those pursuing mathematics degrees. At my alma mater, it was indeed the case that physics students and engineering students were exposed to a different set of course material for foundational courses like linear algebra and complex analysis. Just like compiler theory, if you don't write compilers maybe it's not that useful and you shouldn't be spending too much time on it, but it would be presumptuous to say that delivering a full compiler course is a fundamentally incorrect approach, because somebody has to make that sausage.
BOOSTERHIDROGEN 3y ago
Any books how to read and write proofs ? The above statement of zero vector is unique, I have no idea what is that means.
- navanchauhan 3y ago
  I would recommend Book of Proof: https://www.people.vcu.edu/~rhammack/BookOfProof/ https://www.people.vcu.edu/~rhammack/BookOfProof/ We used this in my Discrete Mathematics class (MATH 2001 @ CU Boulder) (it is a pre-requisite for most math classes). The section about truth tables did overlap a bit with my philosophy class (PHIL 1440 Critical Thinking)
- mananaysiempre 3y ago
  > The above statement of zero vector is unique, I have no idea what is that means. In isolation, nothing. (Neither does the word “vector”, really.) In the context of that book, the idea is more or less as follows: Suppose you are playing a game. That game involves things called “vectors”, which are completely opaque to you. (I’m being serious here. If you’ve encountered about some other thing called “vectors”, forget about it—at least until you get to the examples section, where various ways to implement the game are discussed.) There’s a way to make a new vector given two existing ones (denoted + and called “addition”, but not the same as real-number addition) and a way to make a new vector given an existing one and a real number (denoted by juxtaposition and called “multiplication”, but once again that’s a pun whose usefulness will only become apparent later) (we won’t actually need that one here). The inner workings of these operations in turn are also completely opaque to you. However, the rules of the game tell you that 1. It doesn’t matter in which order you feed your two vectors into the “addition” operation (“add” them): whatever existing vectors v and w you’re holding, the new vector v+w will turn out to be the same as the other new vector w+v. 2. When you “add” two vectors and then “add” the third to the result, you’ll get the exact same thing as when you “add” the first to the “sum” of the second and third; that is, whatever the vectors u, v, and w are, (u+v)+w is equal to u+(v+w). (Why three vectors and not four or five? It turns out that you have the rule for three, you can prove those for four, five, and so on, even though there are going to be many more ways to place the parens there. See Spivak’s “Calculus” for a nice explanation, or if you like compilers, look up “reassociation”.) 3. There is [at least one] vector, call it 0, such that adding it to anything else doesn’t make a difference: for this distinguished 0 and whatever v, v+0 is the same as v. Let’s now pause for a moment and split the last item into two parts. We’ll say a vector u deserves to be called a “zero” if, whatever other vector we take [including u itself!], we will get it back again if we add u to it; that is, for any v we’ll get v+u=v. This is not an additional rule. It doesn’t actually tell us anything. It’s just a label we chose to use. We don’t even know if there are any of those “zeros” around! And now we can restate rule 3, which is a rule: 3. There is [at least one] “zero”. What the remark says is that, given these definitions and the three rules, you can show, without assuming anything else, that there is exactly one “zero”. (OK, what the remark actually says is that you can prove that from the full set of eight rules that the author gives. But that is, frankly, sloppy, because the way rule 4 is phrased actually assumes that the zero is unique: either you need to say that there’s a distinguished zero such that for every v there’s a w with v+w= that zero, or you need to say that for every v there’s a w such that v+w is a zero, possibly a different one for each v. Of course, it doesn’t actually matter!—there can only be one zero even before we get to rule 4. But not making note of that is, again, sloppy. This kind of sloppiness is perfectly acceptable among people who have seen this sort of thing before, say done finite groups or something like that. But if the book is supposed to be give a first impression, this seems like a bad idea. Perhaps a precalculus course of some sort is assumed. Read Spivak, seriously. He’s great. Not linear algebra, though.)
margalabargala 3y ago
That's fine, but then maybe this book shouldn't then be the top comment for being recommended as a "good first course" then?
- Koshkin 3y ago
  Well, perhaps it's not meant to be the first book on mathematics one sees in their life.
  - echion 3y ago
    It's supposed to be the introduction to rigorous proof, so it is supposed to be "the first book on [formal] mathematics" for its audience.
  - margalabargala 3y ago
    From the second paragraph of the introduction to the book we are discussing: > Besides being a first course in linear algebra it is also supposed to be a first course introducing a student to rigorous proof, formal definitions---in short, to the style of modern theoretical (abstract) mathematics. So it's certainly meant to be the first math book one sees in their life that discusses rigorous proofs.
echion 3y ago
> this book is meant for the audience who can read and write proofs It seems like the opposite is true: "It is intended for a student who, while not yet very familiar with abstract reasoning, is willing to study more [than a] "cookbook style" calculus type course." (from the link). If your point is one can't learn linear algebra before learning "abstract [mathematical] reasoning"...don't think you're the main target audience of a subject as practical as linear algebra.
- bscphil 3y ago
  It also goes on to say > Besides being a first course in linear algebra it is also supposed to be a first course introducing a student to rigorous proof, formal definitions---in short, to the style of modern theoretical (abstract) mathematics. So I think it's fair to say that the book (ought to) assume zero knowledge of proofs, contra your parent's claim that the audience is expected to be able to read and write proofs.
itsoktocry 3y ago
"Here's a great first course in Linear Algebra!" "If you can't read and write mathematical proofs, this isn't for you." Math snobs are funny, and are fully to blame for people hating math.
- Sharlin 3y ago
  A first course in linear algebra still assumes background information, because linear algebra is not a basic topic. It’s not meant to be a first course in math. Math builds on itself and it would be incredibly inconvenient if every course everywhere would have to include a recap of basic things. And proofs are among the most fundamental things in math! Programming courses or articles or books, beyond the 101 level, don’t teach you again and again the basics of declaring a variable and writing a loop either! No field does that. Wrt linear algebra in particular, there are plenty of resources aimed at programmers thanks to its relevance in computer graphics and so on. They typically skip proofs and just tell you that this is how matrix multiplication is defined, but they don’t teach you math, merely using math. Which can be plenty enough to an engineer.
- eigenket 3y ago
  This is not snobbery, some subjects just have prerequisites. You can't learn computer science without having a good sense of what an "algorithm" is, you have to know how to read and write and understand algorithms. Similarly you can't learn math without having a good sense of what a proof is, reading, writing and understanding proofs is the heart of what math is. Even more strongly, trying to learn math without a solid understanding of how proofs work is something like studying English literature while refusing to learn how to read English.
  - margalabargala 3y ago
    > Even more strongly, trying to learn math without a solid understanding of how proofs work is something like studying English literature while refusing to learn how to read English. It depends why you're trying to learn math. Are you interested in math for math's sake, or are you trying to actually do something with it? If it's the former, then yeah, you need proofs. Otherwise, like in your analogy, it's like studying English literature without knowing any english grammar rules. But if you're trying to apply the math, if you're studying linear algebra because it's useful rather than for its own sake, then you don't need proofs. To follow the same analogy, it's like learning enough English to be conversational and get around America, without knowing what an "appositive" is. The software industry, similarly, is full of people who make use of computer science concepts, without having rigorously studying computer science. You can't learn true "computer science" without an understanding of discrete math, but you can certainly get a job as an entry-level SWE without one. You don't need discrete math to learn python, see that it's useful, and do something interesting with it. The same applies to linear algebra. Everyone who does vector math doesn't need to be able to prove that the tools they are using work. If everyone who does vector math is re-deriving their math from first principles, then something's gone terribly wrong. There's a menu of known, well-defined treatments that can be applied to vectors, and one can read about them and trust that they work without having proven why they work. EDIT: it occurs to me, an even stronger analogy of this point, is that it is entirely possible to study computer science, without having any understanding of electrical engineering or knowing how a transistor works.

BoiledCabbage 3y ago

> This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. The thing that mathematicians refuse to admit is that they are extremely sloppy with their notation, terminology and rigor. Especially in comparison to the average programmer. They are conceptually/abstractly rigorous, but in "implementation" are incredibly sloppy. But they've been in that world so long they can't really see it / just expect it. And if you debate with one long enough, they'll eventually concede and say something along the lines of "well math evolved being written on paper and conciseness was important so that took priority over those other concerns." And it leaks through into math instruction and general math text writing. Programming is forced to be extremely rigorous at the implementation level simply because what is written must be executed. Now engineering abstraction is extremely conceptually sloppy and if it works it's often deemed "good enough". And math generally is the exact opposite. Even for a simple case, take the number of symbols that have context sensitive meanings and mathematicians. They will use them without declaring which context they are using, and a reader is simply supposed to infer correctly. It's actually somewhat funny because it's not at all how they see themselves.

rq1 3y ago
This is ridiculous. The average computer scientist (not only "programmer", as a js dev would be) never wrote lean/coq or similar, and is not aware of the Curry-Haskell like theorems and their implications.
- thrwayaistartup 3y ago
  I think you entirely missed the point. GP put it well: >> They are conceptually/abstractly rigorous, but in "implementation" are incredibly sloppy. Maturity in concept-space and the ability to reason abstractly can be achieved without the sort of formal rigor required by far less abstract and much more conceptually simple programming. I have seen this first hand TAing and tutoring CS1. I regularly had students who put off their required programming course until senior year. As a result, some were well into graduate-level mathematics and at the top of their class but struggled deeply with the rigor required in implementation. Think about, e.g., missing semi-colons at the end of lines, understanding where a variable is defined, understanding how nested loops work, simple recursion, and so on. Consider something as simple as writing a C/Java program that reads lines from a file, parses them according to a simple format, prints out some accumulated value from the process, and handles common errors appropriately. Programming requires a lot more formal rigor than mathematical proof writing.

ravi-delia 3y ago

Axler serves as an adequate first introduction to linear algebra (though it is intended to be a second, more formal, pass through. Think analysis vs calculus), but it isn't intended to be a first introduction to all of formal mathematics! A necessary prereq is understanding some formal language used in mathematics- what unique means is included in that. Falling entirely back on physical intuition is fine for students who will use linear algebra only in physical contexts, but linear algebra is often a stepping stone towards more general abstract algebra. That's what Axler aims to help with, and with arbitrary (for instance) rings there isn't a nice spacial metaphor to help you. There you need to have developed the skill of looking at a definition and parsing out what an object is from that.

Tazerenix 3y ago

Mathematicians are well aware of complaints like these about introductions to their subjects, by the way. It is for a reason that this book introduces the theory of abstract vector spaces and linear transformations, rather than relying on the crutch of intuition from Euclidean space. If you want to become a serious mathematician (and this is a book for such people, not for people looking for a gentle introduction to linear algebra for the purposes of applications) at some point it is necessary to rip the bandaid of unabstracted thinking off and engage seriously with abstraction as a tool. It is an important and powerful skill to be presented with an abstract definition, only loosely related to concrete structures you have seen before, and work with it. In mathematics this begins with linear algebra, and then with abstract algebra, real analysis and topology, and eventually more advanced subjects like differential geometry. It's difficult to explain to someone whose exposure to serious mathematics is mostly on the periphery that being exposed forcefully to this kind of thinking is a critical step to be able to make great leaps forward in the future. Brilliant developments of mathematics like, for example, the realisation that "space" is an intrinsic concept and geometry may be done without reference to an ambient Euclidean space begin with learning this kind of abstract thinking. It is easy to take for granted the fruits of this abstraction now, after the hard work has already been put in by others to develop it, and think that the best way to learn it is to return back to the concrete and avoid the abstract.

fakecontent000 3y ago
God I just hate you so much.
resource0x 3y ago
>... rather than relying on the crutch of intuition from Euclidean space Euclidean space is not a good crutch, but there are other, much more meaningful, crutches available, like (orthogonal) polynomials, Fourier series etc. Not mentioning any motivations/applications is a pedagogical mistake IMO. I think we need some platform for creating annotated versions of math books (as a community project) - that could really help.
- Tazerenix 3y ago

Ridj48dhsnsh 3y ago

I absolutely agree about additional rigor and precision making math easier to learn. Only after you're familiar with the concepts can you be more lazy. That's the approach taken by my favorite math book: https://people.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf https://people.math.harvard.edu/~shlomo/docs/Advanced_Calcul...

Galanwe 3y ago

> This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. This is _precisely_ the opinion of Roger Godement, French mathematician and member of the Bourbaky group. I would highly recommend his books on Algebra. They are absolutely uncompromising on precision and correctness, while also being intuitive and laying down all the logical foundations of their rigor. Overall, I cannot recommend enough the books of the Bourbaky group (esp. Dieudonne & Godement). They are a work of art in the same sense that TAOCP is for computer science.

ykonstant 3y ago
Unfortunately, some of the Bourbaki books need to be read in French, because the typesetting on the English translations is so atrocious as to be unreadable; as a consolation, the typesetting on the original French is, as always, immaculate.

seanhunter 3y ago

> This is actually why I feel that mathematical texts tend to be not rigorous enough, rather than too rigorous. On the surface the opposite is true - you complain, for instance, that the text jumps immediately into using technical language without any prior introduction or intuition building. My take is that intuition building doesn't need to replace or preface the use of formal precision, but that what is needed is to bridge concepts the student already understands and has intuition for to the new concept that the student is to learn. If you read the book in the original post you may find it's absolutely for you. Axler assumes you know only the real numbers, then starts by introducing the commutative and associative properties and the additive and multiplicative identity of the complex numbers[1]. Then he introduces fields and shows that, hey look we have already proved that the real and complex numbers are fields because we've established exactly the properties required. Then he goes on to multidimensional fields and proves the same properties (commutativity and associativity and identities) in F^n where F is any arbitrary field, so could be either the real or the complex numbers. Then he moves onto vectors and then onto linear maps. It's literally chapter 3 before you see [ ] notation or anything that looks like a matrix, and he introduces the concept of matrices formally in terms of the concepts he has built up piece by piece before. Axler really does a great job (imo) of this kind of bridge building, and it is absolutely rigorous each step of the way. As an example, he (famously) doesn't introduce determinants until the last chapter because he feels they are counterintuitive for most people and you need most of the foundation of linear algebra to understand them properly. So he builds up all of linear algebra fully rigorously without determinants first and then introduces them at the end. [1] eg he proves that there is only one zero and one "one" such that A = 1*A and A = 0 + A.