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.
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?
> taking all the optional extra math courses (including linear algebra), without there ever being a big emphasis on proofs
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.
> 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.
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?
> 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?
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 sibling comment answered most of what you wrote. So, I'll just add that I'm talking about the present day, not 20+ years ago. I don't know about your experience in Australia 20+ years ago, but I'm teaching real students, today, who just got out of high school, in Western Europe. Not hypothetical students 20 years ago in Australia. And based on what the Australian colleagues I met at various conferences told me, their teaching experience in Australia today isn't really different from mine.
FWiW I started out in Engineering and transferred out to a more serious mathematics | physics stream.
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").
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.
I can only speak to my own experiences, but the math courses were not customised for engineering students. I sat next to students who were planning to become mathematicians. Linear Algebra was an optional course for me.
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.
I can second this, in my Engineering degree the Linear Algebra course (and the Calculus course) were both taught by the Math Faculty at my Uni.
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...
Engineers can learn linear algebra from an engineering perspective, i.e. not emphasizing proofs, and that’s fine, but the books being discussed are not intended for that audience.
I don't know whom to agree with. Maybe there need to be two tracks, and it might not even depend on discipline, but just personal preference. Do you love math as an art form, or as a problem solving tool? Or both?
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.
> 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.
Most engineers don’t learn linear algebra, they learn a bit of matrix and vector math in real Euclidean spaces. They don’t learn anything about vector spaces or the algebra behind all of it. What they learn would only be considered “advanced mathematics” 200 years ago, when Gauss first developed matrices.
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?
It covers most of the topics covered in a first course in linear algebra, it’s just very application-specific. It has some basic proofs in the exercises, but nothing overly difficult, and the more involved proofs give you explicit instructions on the steps to take.
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?
This depends on who you ask. Personally, I found studying pure math incredibly rewarding. It gave me the confidence to be able to say that I can look at any piece of mathematics and figure it out if I take the time to do so.
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.
Are there courses/books on "applied linear algebra"? You are right in some sense, but wrong in some sense. Linear algebra at a 100 level without any really deep understanding is still incredibly useful. Graphics (i guess you sort of call this out), machine learning etc.
Most university math curriculums have a clear demarcation between the early computation-oriented classes (calculus, some diff eq.) and later proof-oriented classes. Traditionally, either linear algebra or abstract algebra is used as the first proof-oriented course, but making that transition to proof-based math at the same time as digesting a lot of new subject matter can be brutal, so many schools now have a dedicated transition course (often covering a fair bit of discrete mathematics). But there's still demand for textbooks for a linear algebra course that can serve double-duty of teaching engineering students a bag of tricks and give math students a reasonably thorough treatment of the subject.
>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
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.
There are (at least) two different things we're calling "linear algebra here", roughly speaking one is building the tools and one is using the tools.
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.
I don't think I've ever done more rote manual calculation than for my undergrad linear algebra class! On tests and homework just robotically inverting matrices, adding/subtracting them (I think I even had to do some of that in high school algebra), multiplying them (yuck). It was tedious and frustrating and anything but theoretical.
I've learned linear algebra course quality varies substantially. One acquaintance whom I met after they graduated a big university in Canada reported having to do things like by-hand step-by-step reduced row echelon form computations for 3x4 matrices or larger. I had to do such things in "Algebra 2" in junior high (9th grade), until our teacher kindly showed us how to do the operations on the calculator and stopped demanding work steps. If we had more advanced calculators (he demoed on some school-owned TI-92s, convincing me to ask for a TI-89 Titanium for Christmas) we could use the rref() function to do it all at once.
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.)
Linear algebra is usually a required (or at least strongly encouraged) course for an undergraduate degree in basically any engineering discipline, and it is usually not preceded by a course in "the basics of mathematical proofs".
