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.
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?
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.
Having said that, I’m sure theorem proving was part of it (this was many years ago), I just don’t recall it as being fundamental in any sense. I’m sure that has something more to do with the student than the course work. I liked (and like), maths, but I was there to build my tool chest. A different student, with a different emphasis, would have gotten different things out of the course.
But I think my viewpoint is prevalent in engineering, even from engineers who started with a math degree. The emphasis on “what can I do with this”, relegates theorem proving to annoying busywork.
It could just be me.
The textbook we used was "Linear Algebra: And its Applications" by David C Lay 20 years later I still keep this textbook with me at my desk and consult it a few times a year when I need to jog my memory on something. I consider it to be a very good textbook even if it doesn't it doesn't contain any rigorous proofs or axioms...
The Engineering curriculum as I found it was essentially rote for the first two years.
It had more exams and units than any other courses (including Medicine and Law which tagged in pretty close) and included Chemistry 110 (for Engineers) in the Chemistry Department, Physics 110 (for Engineers) in the Physics Department, Mathematics 110 (for Engineers) in the Mathematics Department, and Tech Drawing, Statics & Dynamics, Electrical Fundementals, etc in the Engineering Department.
All these 110 courses for Engineers covered "the things you need to know to practically use this infomation" .. how to use Linear Algebra to solve loading equations in truss configurations, etc.
These were harder than the 115 and 130 courses that were "{Chemistry | Math | Physics} for Business Majors" etc. that essentially taught familiarity with subjects so you could talk with the Engineers you employed, etc.
But none of the 110 courses got into the meat of their subjects in the same way as the 100 courses, these were taught to instruct people who intended to really master. Maths, Physics, or Chemistry.
Within a week or two of starting first year university I transfered out of the Maths 110 Engineering unit and into Math 100, ditto Chem and Physics. Halfway through second year I formally left Engineering the curriculum altogether (although I later became a professional Engineer .. go figure).
The big distinction between Math 100 V. Math 110 was the 110 course didn't go into how anything "worked", it was entirely about how to use various math concepts to solve specific problems.
Math 100 was fundementals, fundementals, fundementals - how to prove various results, how to derive new versions of old things, etc.
Six months into Math 100 nothing had been taught that could be directly used to solve problems already covered in Math 110.
Six months and one week into Math 100 and suddenly you could derive for yourself from first principals everything required to be memorised in Math 110 and Math 210 (Second year "mathematics for engineers").
This book is for people who want to be mathematicians.
> 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.
Engineers frequently need to learn some fairly advanced mathematics. Are you suggesting they can’t use the same textbooks?
By the way; I don’t think the original poster is wrong as such (every similar textbook is undoubtedly full of proofs), I’m just suggesting a different viewpoint. Not everyone learning Linear Algebra is intending to become a mathematician.
You can see with a quick skim that the content is very application focused. I just don’t know enough to know what I don’t know. If one were to learn Linear Algebra using this textbook, would it be a proper base? Would you have grasped the fundamentals?
There is a chapter on abstract vector spaces and there are a few examples given besides the usual R^n (polynomials, sequences, functions) but there is almost no time spent on these. There is also no mention of the fact that the scalars of a vector space need not be real numbers; that you can define vector spaces over any number field.
There is only a passing discussion of complex numbers (as possible Eigenvalues and in an appendix) but no mention of the fact that vector spaces over the field of complex numbers exist and have an even more well-developed theory than for real numbers.
But more fundamental than a laundry list of missing or unnecessary topics is the fact that it’s application focused. Pure mathematics courses are proof and theory focused. So they cover all the same (and more) topics in much richer theoretical detail, and they teach you how to prove statements. Pure math students don’t just learn how to write proofs in one or two courses and then move on; all of the courses they take are heavily proof-based. Writing proofs, like programming, is a muscle that benefits from continued exercise.
So if you’re studying (or previously studied) science or engineering and learned all your math from that track, switching to pure math involves a bunch of catch up. I’ve met plenty of people who successfully made the switch, but it took a concerted effort.
There seems to be a fundamental difference in mindset between the “applications” based learning of mathematics, and this pure math based version. Are there benefits to be had for a person that only intends to use mathematics in an applied fashion?
I can't speak for those who have only studied math at an applied level directly. My impression of them (as an outsider but also a math tutor) is that they are fairly comfortable with the math they have been using for a while but always find new mathematics daunting.
I have heard famous mathematicians describe this phenomenon as "mathematical maturity" but I don't know if this has been studied as a real social/educational phenomenon.
I went back and forth. I was good at problem solving, but proofs were what made math come alive for me, and I started college as a math major. Then I added a physics major, with its emphasis on problem solving. But I would have struggled with memorizing formulas if I didn't know how they were related to one another.
Today, K-12 math is taught almost exclusively as problem-solving. This might or might not be a realistic view of math. On the one hand, very few students are going to become mathematicians, though they should at least be given a chance. On the other hand, most of them are not going to use their school math beyond college, yet math is an obstacle for admission into some potentially lucrative careers.
At my workplace, there's some math work to be done, but only enough to entertain a tiny handful of "math people," seemingly unrelated to their actual specialty.
Sorry to break it to you, but you didn't take math classes. You took classes of the discipline taught in high school under the homonymous name "math". There is a big difference.
It's the same difference as there is between what you get taught in grade school under the name "English" (or whatever is the dominant language where you live): the alphabet, spelling, pronunciation, basic sentence structure... And what gets taught in high school under the name "English": how to write essays, critically analyze pieces of literature, etc. The two sets of skills are almost completely unrelated. The first is a prerequisite for the second (how can you write an essay if you can't write at all?), so somehow the two got the same name. But nobody believes that winning a spelling bee is the same type of skill as writing a novel.
I know it's a shock to everyone who enters a university math course after high school. Many of my 1st year students are confounded about the fact that they'll be graded on their ability to prove things. They expect the equivalent of cooking recipes to invert matrices, compute a GCD, solve a quadratic equation, or whatever, and balk at anything else. I want them to understand logical reasoning, abstract concepts, and the difference between "I'm pretty sure" and "this is an absolute truth". There's a world of difference, and most have to wait a few years to develop enough maturity to finally get it.
If you look at the comments below, you’ll see that this can’t be strictly true. At least, not 20+ years ago in Australia when I was a student. Some of the courses I took were in the math faculty with students who were going on to become mathematicians. At that time this would have been a quarter load of a semester, and was titled “Linear Algebra”, but I can’t remember if it was 1st/2nd or even 3rd year subject (it’s been too long).
Perhaps the lack of emphasis on proofs (I am not saying proofs were absent, I made another comment with more explanation), was a combination of these being introductory courses, the universities knowledge that there were more than just math faculty students taking them, or changes with time in how the pedagogy has evolved.
What is more interesting to me, is what do you think a student misses out on, from a capability point of view, with an applications focused learning as opposed to one focused on reading and writing proofs?
Would a student who is not intending to become a mathematician still benefit from this approach? Would a middle aged man who was taught some “Linear Algebra” benefit from picking up a book such as the one referenced here?
The generalizable value is not so much in collecting a bunch of discrete capabilities (they're there, but generally somewhat domain-specific) as it is in developing certain intuitions and habits of thought: what mathematicians call "mathematical maturity". A few examples:
- Correcting trivial errors in otherwise correct arguments on the fly instead of getting hung up on them (as demonstrated all over this comment section).
- Thinking in terms of your domain rather than however you happen to be choosing to represent it at the moment. This is why math papers can be riddled with "syntax errors" and yet still reach the right conclusions for the right reasons. These sorts of errors don't propagate out of control because they're not propagated at all: line N+1 isn't derived from line N: conceptual step N+1 is derived from conceptual step N, and then they're translated into lines N+1 and N independently.
- Tracking, as you reason through something, whether your intuitions and heuristics can be formalized without having to actually do so.
- More generally, being able to fluently move between different levels of formality as needed without suffering too much cognitive load at the transitions.
- Approaching new topics by looking for structures you already understand, instead of trying to build everything up from primitives every time. Good programmers do the same, but often fail to generalize it beyond code.
> Would a student who is not intending to become a mathematician still benefit from this approach?
If they intend to go into a technical field, absolutely.
> Would a middle aged man who was taught some “Linear Algebra” benefit from picking up a book such as the one referenced here?
Depends on what you're looking for. If you want to learn other areas of math, linear algebra is more or less a hard requirement. If you want to be able to semiformally reason about linear algebra faster and more accurately, yes. If you just want better computational tricks, drink deep or not at all: they're out there, but a fair bit further down the road.
The Linear Algebra chapter links to 'longer' courses the chapter is based on:
>Pavel Grinfeld’s series on linear algebra: http://tinyurl. com/nahclwm
>Gilbert Strang’s course on linear algebra: http://tinyurl. com/29p5q8j
>3Blue1Brownseries on linear algebra: https://tinyurl. com/h5g4kps
That's quite far from what the author of the book you're discussing wrote:
> [this book/course] is also supposed to be a first course introducing a student to rigorous proof, formal definitions
I can't write a proof to save my life, but I'm going to keep using linear algebra to solve problems and make money, nearly every day. Sorry!
We had this discussion about Data Science years ago: "you aren't a real Data Scientist unless you fully understand subjects X, Y, Z!"
Now companies are filled to the brim with Data Scientists who can't solve a business problem to save their life, and the companies are regretting the hires. Nobody cares what proofs they can write.
The mathematicians need to understand the basics of of mathematical proofs to learn how to prove new interesting (and sometimes useful) stuff in linear algebra. You have to do the math stuff in order to come up with some new matrix decomposition or whatever.
The engineers/data scientists/whatever people just need to understand how to use them.
You don't need to know how to build a car to drive one. The mathematicians are building the cars, you're using them.
In my actual linear algebra class in freshman year college we were introduced to a lot of proper things I wish I had seen before, along with some proofs but it wasn't proof heavy. I did send a random email to my old 9th grade teacher about at least introducing the concept of the co-domain, not just domain and range, but it was received poorly. Oh well. (There was a more advanced linear algebra class but it was not required for my side. The only required math course that I'd say was proof heavy was Discrete Math. An optional course, Combinatorial Game Theory, was pretty proof heavy.)
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.
> 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.
> 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.
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)
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.)
Did you mean Spivak, Michael [1]?
A vector space is defined as having a zero vector, that is, a vector v such that for any other vector w, v + w = w.
Saying the zero vector is unique means that only one vector has that property, which we can prove as follows. Assume that v and v’ are zero vectors. Then v + v’ = v’ (because v is a zero vector). But also, v + v’ = v’ + v = v, where the first equality holds because addition in a vector space is commutative, and the second because v’ is a zero vector. Since v’ + v = v’ and v’ + v = v, v’ = v.
We have shown that any two zero vectors in a vector space are in fact the same, and therefore that there is actually only one unique zero vector per vector space.
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.
"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.
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.
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.
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.
Sure, but then you're not studying math, you're studying applications of math or perhaps you're even studying some other subject like engineering which is built on top of applications of math.
To add an extra analogy to the pile, its like learning to drive a car vs learning how to build a car. Sure, its completely valid to learn how to drive a car without knowing how to build one, but no one says they're learning automotive engineering when they're studying for their driving test. Its a different subject.