Linear Algebra is harder than Calculus

[homeomorphic]

I think he's putting in much more effort than you are giving him credit for...

No one said anything about effort. We said something about using a bad strategy to learn. I don't see how the stuff you put in bold was anything but negative. In fact, it's mostly the stuff I would put in bold to show how he was going about it the wrong way. Khan and Strang's lectures may well be bad for him, but probably less so for other people, though I have no opinion on them, since I have not seen them. I used Strang's linear algebra book when I first learned it, and I'm not sure what I think of it at this point, since that was before I had figured out how to do math. I would say it's not terrible, but not as much intuition there as I would put if I wrote a linear algebra book.

Plus you also have to keep in mind that for about 20 years I did basically no math. I pretty much just gave up on precalculus when I was in high school. I remember being so bad that I actually thought in f(x) that f was some sort of variable.

This is one reason why I don't like the idea of traditional classes, and especially lectures. They are one size fits all. The ideal is tutoring one or two students at a time or some equivalent thing (going to office hours can give you a limited amount of this). I think most students aren't going to have too much difficulty with the basic "knowing how to do the basics" part. They might not be great at math, but I would expect them not to have too much trouble with basic algebra, like beginning calculus students would. However, I might expect them to still be somewhat conceptually impaired, which is part of the problem with books directed at that sort of audience. At any rate, the point is that if someone who has trouble with math wants to learn linear algebra, there SHOULD be resources out there for them to learn. But they have to realize that if they don't have a better mastery of the prereqs, the class might be above their level because the target audience (being, perhaps, the average student who winds up in the class) is above their level.

 

[YAHA]

Yep its true. Linear algebra is ridiculously hard and it is made harder by bad teachers.

However there's nothing you can do about it until you become a professor of mathematics and write your own book on the subject so you have to suck it up to pass the class.

Dude, if you consider Strang a "bad teacher" then I am not sure what you are looking for. I realize that certain professors will speak better to you than others but calling Strang a bad teacher? I took LA just under a year ago. I used a typical text (cant even recall the author). I personally found it too shaky and not rigorous enough. Thus, I obtained one by Shilov (much harder and abstract) and was thouroughly rewarded for that. Additionally, I supplemented the class by watching the lectures of the "bad teacher". Needless to say, I found them marvelous.

I agree that LA is more abstract than calculus. In your case, this should read that LA is less susceptible to students simply drilling the techniques and thereby "passing the class" rather then understanding material deeply.

 

[chill_factor]

Dude, if you consider Strang a "bad teacher" then I am not sure what you are looking for. I realize that certain professors will speak better to you than others but calling Strang a bad teacher? I took LA just under a year ago. I used a typical text (cant even recall the author). I personally found it too shaky and not rigorous enough. Thus, I obtained one by Shilov (much harder and abstract) and was thouroughly rewarded for that. Additionally, I supplemented the class by watching the lectures of the "bad teacher". Needless to say, I found them marvelous.

I agree that LA is more abstract than calculus. In your case, this should read that LA is less susceptible to students simply drilling the techniques and thereby "passing the class" rather then understanding material deeply.

I am not talking about the writer of the book, but about teachers assigned to linear algebra in general.

 

[Windowmaker]

This is called "letting someone else do your thinking for you.



As is this.



It sounds more like you managed to get through Calc without learning how to tackle problems on your own - without solution manuals and without youtube - and now you're paying for it.

He is insinuating that the videos are doing the work for him which can be interpreted as lacking effort...

 

[20Tauri]

I agree that LA is more abstract than calculus. In your case, this should read that LA is less susceptible to students simply drilling the techniques and thereby "passing the class" rather then understanding material deeply.

Yeah, that's how I understood it.

Also, here's my two cents on the "bad professors teaching linear algebra" thing. At my school, anyway, you were fairly likely to get a professor who'd scare you into learning things--kind of a sink or swim situation. I wouldn't say that we worried about getting bad professors so much as getting one who would expect quite a lot from us. Maybe frightening your students is bad pedagogy, or maybe it's a kindness to both the good and the bad students--you either learn rigor early, or you get the chance to cut out while you still have time to choose another major. I don't know; I got "lucky" in LA as in most of my classes and had a professor who was both a good teacher and super nice.

 

[g.lemaitre]

No one said anything about effort. We said something about using a bad strategy to learn. I don't see how the stuff you put in bold was anything but negative. In fact, it's mostly the stuff I would put in bold to show how he was going about it the wrong way.

Right, great advice: don't watch lectures, don't switch textbooks when they don't work for you, and don't consult manuals that show you how to do a problem when you can't figure it out yourself. In a word what you're saying is: if you can't understand something, understand it anyway.

In any case, I went back to Richard Hill's textbook and I'm more or less cruising now.

 

[gravenewworld]

I found LA to be much easier than Calc, especially Calc II. Calc II was one of the toughest math classes I've ever taken. Many times I feel instructors take ridiculous integrations, and reverse engineer them by using a computer program.

 

[g.lemaitre]

I found LA to be much easier than Calc, especially Calc II. Calc II was one of the toughest math classes I've ever taken. Many times I feel instructors take ridiculous integrations, and reverse engineer them by using a computer program.

I think LA depends a lot on what textbook you're using, Calculus too. For instance, if I wanted to rate the difficulty of math texbooks it would be as follows (10 being the hardest, 7 or over means I can't really learn anything):
james stewart, calculus - 8
thomas, calculus - 5
strang, linear algebra - 10
hill, linear algebra - 4
carlen, linear algebra - 7
videos are much easier
krista king's calc lectures - 1
patrick jmt's calc lectures - 1
strang's linear algebra lectures - 7
sal khan's linear algebra lectures - 5

 

[homeomorphic]

Right, great advice: don't watch lectures, don't switch textbooks when they don't work for you, and don't consult manuals that show you how to do a problem when you can't figure it out yourself. In a word what you're saying is: if you can't understand something, understand it anyway.

You are totally putting words in our mouths. No one said don't watch lectures or don't switch textbooks (though some people said not to watch youtube videos, I disagree with that, just don't watch the ones that do everything for you). In fact, switching textbooks is exactly what I would tell you to do. The only thing we were saying was not to rely too much on worked examples. It's not about memorizing procedures that have no meaning to you, just you can get the answer. Your standards of "not being able to figure it out" are probably extremely low, I would suspect. As a grad student, when I do problems, I probably spend most of my time being stuck and not knowing how to figure it out myself. However, I keep thinking about it, and given enough time, I can usually figure it out.

How about what I actually told you to do, instead of what you made up that I told you to?

If you read what I said, I basically said "get a tutor". Barring financial issues, that is the ideal thing for this scenario, rather than worked solutions.

In any
case, I went back to Richard Hill's textbook and I'm more or less cruising now.

Glad you found something that worked for you. However, if you are just memorizing procedures or not understanding things deeply, it could possibly be an illusion that it is working for you. Not saying it is, just saying it could be if that's the way you try to learn.

 

[g.lemaitre]

The only thing we were saying was not to rely too much on worked examples. It's not about memorizing procedures that have no meaning to you, just you can get the answer. Your standards of "not being able to figure it out" are probably extremely low, I would suspect. As a grad student, when I do problems, I probably spend most of my time being stuck and not knowing how to figure it out myself. However, I keep thinking about it, and given enough time, I can usually figure it out.

you're right. my previous comment was a bit impulsive. if I were a young undergrad or grad looking to work in theoretical physics or experimental physics since i think there are so few jobs in theoretical physics i would do things a lot differently. Right now, I just want to get through QM then I want to get back to philosophy and literature. (I realize as a math person you probably think I'm nuts for enjoy phil and lit) Then I have to actually get a real job since my current job only pays 11 dollars an hour. However, I do have ambitions to eventually get to the point where I can understand this equation before I turn 60:

https://www.physicsforums.com/showthread.php?t=622391

So when I make my next big attempt to understand mathematical physics I am going to spend more time on trying to develop some real mathematical tools rather than the fake ones that I have now. I'm going to spend more time studying math in a much more systematic way and also spending a lot of time on hard problems that take about 3 hours to solve solely as an exercise to train my mind to think creatively. right now, i simply don't have the time. my strength is in the humanities and currently i have accomplished very little in the humanities that would make a name for myself. however, I'm very worried that when i actually start a family i probably won't have time for any kind of independent study. if you want to learn QED and all that you've got to study it 40 hours a week and it might be a very long time before i get that kind of time again. i have that time now since my job has almost no demands but it won't last forever.

 

[homeomorphic]

Right now, I just want to get through QM then I want to get back to philosophy and literature. (I realize as a math person you probably think I'm nuts for enjoy phil and lit)

No, philosophy addresses interesting questions. I'm not sure about formally studying what all the philosophers said, but I'm sure there's something to be gained there. Literature is not bad, I just never have time for it, nor do I understand it very well.

So when I make my next big attempt to understand mathematical physics I am going to spend more time on trying to develop some real mathematical tools rather than the fake ones that I have now. I'm going to spend more time studying math in a much more systematic way and also spending a lot of time on hard problems that take about 3 hours to solve solely as an exercise to train my mind to think creatively.

A hard graduate level problem might take a few days to solve, just to give you an idea, and it is meant for people who are fairly good at math. My thesis problem is taking me 3-4 years to solve, partly because I'm not very good at managing big projects like that.

however, I'm very worried that when i actually start a family i probably won't have time for any kind of independent study. if you want to learn QED and all that you've got to study it 40 hours a week and it might be a very long time before i get that kind of time again. i have that time now since my job has almost no demands but it won't last forever.

I'm not sure if you have to study 40 hours a week. Maybe if you just did 20 hours a week, it would just take you twice as long, 10 hours a week, 4 times as long, etc (the trend might not actually be linear, but the idea still works to some extent).

 

[clope023]

If you read what I said, I basically said "get a tutor". Barring financial issues, that is the ideal thing for this scenario, rather than worked solutions.

I realize old physicists and mathematicians tend to be stubborn about this but switching textbooks along with looking at different worked out solutions did more for my understanding of material than tutors ever did. More often than not its the procedure of approaching problems that has students stuck, and so solution books like the schaum's outlines were a big help in my understanding of how theory is applied and after enough practice with worked out solutions I can look at problems and solve them without help. This either or mentality you guys have about worked out solutions is leaving out the middle ground.

 

[homeomorphic]

I realize old physicists and mathematicians tend to be stubborn about this but switching textbooks along with looking at different worked out solutions did more for my understanding of material than tutors ever did.

I'm a young mathematician, not old, and I am also a tutor. I have been saying, don't rely on worked solutions too much. I personally never needed them, but if they are used very sparingly and only when you are really stuck, it probably won't hurt. A tutor will usually guide you through problems by asking you questions and giving you hints, rather than just giving you the answer. I don't know of any tutor who would just give someone a worked solution. If they just give you the answers, they are not doing a good job, unless maybe it's just some trick that they need where it doesn't really matter that much if they come up with it themselves.

More often than not its the procedure of approaching problems that has students stuck, and so solution books like the schaum's outlines were a big help in my understanding of how theory is applied and after enough practice with worked out solutions I can look at problems and solve them without help.

As long as you aren't parroting solutions, that is the thing to avoid. You can learn to parrot solutions and as long as no one throws you any curve balls, you might be able to solve problems without help after some practice. But you need to also be able to handle problems that aren't exactly like the examples you have seen, and you need to be able to know when your solution is actually correct.

This either or mentality you guys have about worked out solutions is leaving out the middle ground.

I've just been saying don't rely on them too much because that leads to just memorizing procedures, plus lack of practice in thinking for yourself.

 

[homeomorphic]

Also, maybe students sometimes get stuck on procedures because they don't even care about the concepts or aren't even aware that the concepts exist. These concepts would help them solve the problems, but since they hate concepts, they would prefer to rely on parroting solutions. They think the solution is just to copy solutions, rather than to understand the theory, but they don't realize that understanding the theory would help a lot with the solutions.

 

[johnqwertyful]

At my JC, linear algebra teachers were notoriously bad. My teacher was absolutely horrible. Didn't understand the subject himself. The second or third day of class, this other student and I had to explain to him the difference between REF and RREF. He was teaching it wrongly (well, the book contradicted him), and we called him out on it. He didn't understand it.

One question we had on a test was "what is the minimum and maximum norm of <2,3>?" or some other single constant vector. He didn't understand why the students were having trouble with it for like 20 minutes. Until he just asked us to find the norm. Which was better.

Also, for the first like 15 weeks, we covered chapters 1-4. The last week we covered 5-8. We did half the semester in the last week.

 

[member 392791]

yeah best not to go to lecture either, or talk to your professor for help. You better go at this one alone, rather than utilize the sources that were never available before

 

[g.lemaitre]

At my JC, linear algebra teachers were notoriously bad. My teacher was absolutely horrible. Didn't understand the subject himself. The second or third day of class, this other student and I had to explain to him the difference between REF and RREF. He was teaching it wrongly (well, the book contradicted him), and we called him out on it. He didn't understand it.

Truly amazing. I guess mathematics is one field where you can get a professorship. In philosophy for each position that opens there are 800 applications although admittedly a strong majority of them are obviously unqualified.

 

[20Tauri]

Truly amazing. I guess mathematics is one field where you can get a professorship. In philosophy for each position that opens there are 800 applications although admittedly a strong majority of them are obviously unqualified.

Well, in a junior college setting, you will probably have people with master's degrees teaching at least some of the classes. Additionally, some people teach junior/community college on a part-time or adjunct basis, so they may not have "professorships," per se, even though they are professors. Not to say that either of these things makes them unqualified--I only mean that the professor in question might not have managed to obtain a PhD and tenure while not understanding linear algebra.

Of course I know nothing of johnqwertyful's particular professor and situation, so if I am at sea here, please do ignore me.

 

[Sankaku]

My first encounter with LA was of the "throw you in the deep end" variety. It was rough, but I now feel it is one of the most beautiful areas of mathematics.

One problem I see is that intro LA classes come in two varieties. The "applied" LA courses for engineers and physicists try to cram dozens of applied tools into a short time and skip the understanding of what is going on. Courses for mathematicians tend to jump right into proofs and, if you aren't ready, will leave you behind very quickly. If you want to enjoy LA, take a solid intro proof course (something like Discrete math is good) and then take a proof-based course for mathematicians.

I realize as a math person you probably think I'm nuts for enjoy phil and lit

You might want to be cautious about throwing generalizations around. Real people are not stereotypes and you may find that some people you meet (and talk to on the internet) have a strong background in both the sciences and the humanities.

I guess mathematics is one field where you can get a professorship.

From one comment about a junior college you are able to deduce this? Again, I suggest you do a little research before jumping to conclusions.

 

[mathwonk]

Linear of course is a much easier subject, both to use and to understand, than calculus. In fact differential calculus is precisely the science of how to use the infinitely easier subject of linear algebra locally to approximate non linear phenomena.

This question seems to be about which course is easier, and it might be rephrased "thinking is harder than following rules mindlessly". If linear algebra were taught comparably to a cookbook calculus course, it would consist only of row reducing 2 by 2, or possibly 3 by 3, matrices. Definitions of concepts such as eigenvectors would be omitted.I.e. unfortunately for the student who has never seen a real math course, linear algebra is so much easier inherently than calculus, that it is often used as the first course in learning to think abstractly. i.e. the content is so simple that a deeper approach is taken to learning it.

It is also often the first course in which higher dimensions are introduced geometrically.

So it is true that a linear algebra course, as the first abstract course, is often a hard experience. However a real calculus course, had it been offered, would have seemed much much harder.

I recommend a well written introductory linear algebra book, like that by Paul Shields, which teaches the ideas restricted to 2 and 3 dimensions, and clearly as a first step.here is a used one for $4.

http://www.abebooks.com/servlet/SearchResults?an=paul+shields&sts=t&tn=elementary+linear+algebra

or if you prefer, here is a new one for $120.

https://www.amazon.com/dp/0879011211/?tag=pfamazon01-20This is not a mickey mouse book, as it was written by a professor at stanford. i.w. do not plunge into a book like that of hoffman and kunze, if you think linear algebra is hard.