Linear Algebra is harder than Calculus

[g.lemaitre]

But not because LA is intrinsically more difficult than calc but because the pedagological tools for LA are seriously flawed. For calc you just put in the title of the problem you're working on in youtube and out pops about 10 videos that will walk you through how to do the problem. Plus the solution manual I had for calc was honestly easier to understand than the worked examples in the book, it had more steps. For linear Algebra you don't have that. There are only 2 lecture series on LA by Strang and Khan and they aren't all that good. I'm on my 3rd LA text and they've all been pretty bad. The mistake they make is they assume the real basic of LA are too easy and they jump immediately to these abstruse problems that no one cares about. They're almost like brain teasers. I can't even master the mechanics of LA or the notation and then they shove these hard problems in my face without teaching me how to do the real easy stuff.

 

[Number Nine]

Is this a question?
Strang's text/class is far from the best organized on the market, but it's probably one of the most concrete and applications oriented. If you find it "abstruse" and disconnected from the "real world", then you're pretty out of luck as far as LA (or any higher math) goes. There are clearer and more insightful texts, but they're considerably more abstract. Linear algebra is a beautiful subject, but it only gets more abstract from here. Doing LA means doing proofs, which means not having fixed algorithms; solving problems is going to require some amount of exploration.

 

[AlephZero]

"Linear algbra vs calculus" isn't really a fair comparison, any more than comparing say arithmetic with trigonometry.

IMO You should really be comparing linear algebra with analysis, not calculus.

 

[micromass]

It's really not a good thing to rely on youtube videos and solution videos. The further you go in your education, the less such things are available. If you rely too much on these resources, then I think you're studying wrong.

 

[Vanadium 50]

For calc you just put in the title of the problem you're working on in youtube and out pops about 10 videos that will walk you through how to do the problem.

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

Plus the solution manual I had for calc .

As is this.

The mistake they make is they assume the real basic of LA are too easy and they jump immediately to these abstruse problems that no one cares about.

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.

 

[Robert1986]

Also, with LA, this is probably your first course where you are dealing more with the properties of the structures you are studying than with the actual things that make up those structure. This is another layer of abstraction that you have not yet encountered, and so it will seem odd at first.

 

[chill_factor]

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.

 

[Benn]

Think of it as an opportunity! For (what appears to be) the first time, you're being forced to learn something--not simply memorize a youtuber's computation.

 

[chill_factor]

Think of it as an opportunity! For (what appears to be) the first time, you're being forced to learn something--not simply memorize a youtuber's computation.

Anyone that does not like or work in pure mathematics has little need of proving mathematical theorms, only with being skillful at the tools of computing things and arithmetic.

Even in physics, they call things "derivations" and not "proofs" for a reason: it is vastly different than an actual proof in actual math.

 

[g.lemaitre]

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

See, I was under a different impression. Gilbert Strang says in his preface that LA is easier than Calc. So you're saying he's wrong?

Right now, I have to make a critical decision. When I try out the exercises I literally get about 90% of them wrong. Either I'm going to have to go back to the beginning and write down every single thing I understand and ask questions about every single thing I don't understand or just slog through it with only minimal understanding and hope I can get through QM without it. I do self-study and I'm not looking for a job in the science. In fact, I actually belong to the humanities, but I discovered a long time ago that if you stay isolated in the humanities as 97% of humanities majors do, then you're doing yourself an immense disservice and you're setting yourself up for a lopsided understanding of the world.

 

[alan2]

I always get confused by the level of course that people are talking about. If you've never seen any linear algebra in your life and you've only had calculus then I assume you are referring to a course similar that usually taught to freshmen and sophomores in the US which is just matrix and vector algebra in Euclidian space, mostly 3-dimensional. If that's the case you should work through a book like Leon before you try to study general theory in abstract spaces.

 

[genericusrnme]

Linear Algebra is harder than calculus only in the fact that linear algebra is rigorous and elementary calculus is, for the most part, based on intuition

Analysis, which is pretty much just rigorous caluclus is, and I think everyone would agree, harder than linear algebra - imo as far as branches of maths go linear algebra is probably one of the nicer ones, I believe Strang said 'linear algebra is like analysis but everything is behaving nicely'

Perhaps you should learn some set theory and group theory before you start on linear algebra? That way you'll see how you build linear algebra up, starting with sets, adding a set of operations on the set, then adding an action of another set, called a field, on the set.
Or maybe try reading a basic proofs book, like 'How to Prove it - A Structured Approach', that might give you some kind of insight into how to do proofs in linear algebra and introduce you to a level of abstraction beyond the 'intuitive feel' that you're used to.

Either I'm going to have to go back to the beginning and write down every single thing I understand and ask questions about every single thing I don't understand

This is a good approach, if you know how to prove results then slowly they'll seep into you as if they were obvious facts

or just slog through it with only minimal understanding and hope I can get through QM without it.

Quantum Mechanics is pretty tricky on it own, without a solid background in linear algebra you're going to get lost before you even start. You'll also want to understand some basic analysis facts here too.

I'll also recommend Gilbert Strangs "Introduction to Linear Algebra" if you haven't already read it.

The mistake they make is they assume the real basic of LA are too easy and they jump immediately to these abstruse problems that no one cares about

You'll care about those things one day, I think most people felt the same way about the isomorphism theorems when they first saw them but later on realized that it's quite a handy little fact to have with you. (I think a lot of authors don't point out enough when a theorem is going to turn out to be especially useful later on, so you end up wading through a sea of theorems and proofs without any idea why these are important, which is why it's a good idea to work with them for a while and try and apply them to a problem even if it is taylor made to require the use of your newly discovered theorem)

I can't even master the mechanics of LA or the notation and then they shove these hard problems in my face without teaching me how to do the real easy stuff.

If you're having trouble understanding notation then you just need to work with it for a while really. Usually when I start on a new textbook and the author introduces a whole new set of notations I need to stop for a second and look at the definitions, absorb what they mean and perhaps try and make a few examples.

I also agree 100% with Vandium50 and micromass, you've got to stop relying on there being numerous sources (especially lectures online, textbooks imo are far superior to any lecture you haven't personally attended) and you need to know how to derive the results given in the textbook/lectures.

 

[chill_factor]

See, I was under a different impression. Gilbert Strang says in his preface that LA is easier than Calc. So you're saying he's wrong?

Right now, I have to make a critical decision. When I try out the exercises I literally get about 90% of them wrong. Either I'm going to have to go back to the beginning and write down every single thing I understand and ask questions about every single thing I don't understand or just slog through it with only minimal understanding and hope I can get through QM without it. I do self-study and I'm not looking for a job in the science. In fact, I actually belong to the humanities, but I discovered a long time ago that if you stay isolated in the humanities as 97% of humanities majors do, then you're doing yourself an immense disservice and you're setting yourself up for a lopsided understanding of the world.

You don't need to be a genius in linear algebra proofs to do quantum mechanics problems. To fully understand the theory, you might need to go deeper, but I find that its a waste of time to try and "intuitively" understand the matrix formulation since wave mechanics is soooo much easier to visualize, and matrix formulation is in my opinion strictly used for problem solving, especially for spin systems since wave mechanics isn't so nice with spin systems.

In problem solving, you will need to know how to do matrix algebra with square matricies i.e. take eigenvalues, then find eigenfunctions. Its not computationally difficult, just be careful. You'll also need to know the concepts of operator algebra. Pick up a copy of Griffith and try some problems out. If you are interested in *science* as opposed to *math* I think its much better to just straight up do the physics, and pick up whatever math you need along the way, as long as you have a solid math background of: mastery of calculus and multivariable/vector calculus, familiarity with ODEs, integral transforms and basic matrix algebra, and basic understanding of complex variables.

And no, linear algebra is far harder than calculus. Calculus can be "seen" geometrically and its easier to convince yourself that its right. LA cannot. Calculus also doesn't have too many proofs and the formalism is easier to understand since it uses nice familiar things y=f(x) rather than big scary matrices.

Perhaps you should learn some set theory and group theory before you start on linear algebra?

I had an introduction to group theory as part of my molecular spectroscopy class. That stuff is ridiculously difficult, and I was amazed it actually had physical applications. Nonetheless, I do not think it is helpful to learn group theory for elementary linear algebra (the type based on square matrix computations).

 

[Robert1986]

Perhaps you should learn some set theory and group theory before you start on linear algebra? That way you'll see how you build linear algebra up, starting with sets, adding a set of operations on the set, then adding an action of another set, called a field, on the set.
Or maybe try reading a basic proofs book, like 'How to Prove it - A Structured Approach', that might give you some kind of insight into how to do proofs in linear algebra and introduce you to a level of abstraction beyond the 'intuitive feel' that you're used to.

I'm sorry, but I disagree with this advice. You are suggesting that he 1)learns what a group is 2)learns what a field is 3)learns how a field acts on group 4)understand linear algebra.

This is nearly impossible for a few reasons.

First, he is struggling with the abstractness of linear algebra right now. I don't see how adding another layer of abstraction will make things any better.

Second, nearly every algebra book I have seen uses examples from elementary linear algebra (which is what he s doing) as concrete examples. Yes, linear algebra is built from the stuff of Abstract Algebra, but this does not mean that someone needs to understand abstract algebra to understand linear algebra.

What you are suggesting is rather like suggesting that a struggling calc student work his way through baby Rudin. It just doesn't work because the textbook authors assume familiarity with more elementary math.

 

[g.lemaitre]

I guess I wasn't prepared for LA being harder than calc. I didn't take LA that seriously because I thought it would be a breeze, just as calc was more or less a breeze. I was a little caught off guard when I failed to understand it.

 

[deluks917]

If the questions involve proofs the difficulty is more a function of the professor. It is very easy to ask very difficult questions in either subject.

I think most calc textbooks have harder computational questions than LA textbooks.

 

[Muphrid]

Something I haven't seen anyone ask is what aspects of linear algebra in particular you find difficult. That might help tailor any recommendations as far as alternative texts or materials, also.

 

[tamtam402]

Do you happen to speak french? Your name sounds french.

If so, I found the book "Vecteurs, matrices et nombres complexes" by Vincent Papillon to be an excellent book to learn the subject from scratch.

 

[-Dragoon-]

Having taken both calc and linear algebra, I struggled with the latter much more until I started using a better studying strategy. In the long run, I am inclined to say linear algebra is earier than elementary calculus. In calculus, you can get by without understanding the intuition behind theorems and just memorizing algorithms, but that won't work too well in linear algebra unless it is taught to engineers. I found actually going through the theorems one by one with TA's, professor, or other students who were well grounded in the book is all I really needed. Once I understood the theorems, I could answer the most difficult questions on the exams. In calculus, I felt that wasn't the case and there still computational questions that could stump you even if you have a deep, rigorous understanding of the material.

Don't approach linear algebra the same way you approach elementary calculus. It just won't work out too well in most cases. And if you think its hard now, wait until you get to linear transformations/mappings and the relationship between those concepts and what you have learned earlier in the course (rank, nullity, orthogonal projection, etc).

 

[g.lemaitre]

Do you happen to speak french?

Georges Lemaitre was one of the founders of Big Bang theory. Sometimes his name gets attached onto the Friedman-Walker-Robertson-Lemaitre Metric


As for what aspects of LA I'm having trouble with, practically everything except for the real easy arithmetic parts, liking multiplying two matrices together. I've put a few specific questions in the homework section.

 

[Muphrid]

Admittedly, I only found three questions--vector addition, matrix times its transpose, and computing projections--but it seems to me like for you, some aspects of the notation are not natural yet. For instance, the idea that a scalar times a vector multiplies all the elements of that vector. Well, maybe the idea makes sense on paper, but otherwise, the problems about vectors that you posted about have more to do with the arithmetic underlying the calculation than the concept itself.

Sorry I can't be of more help. Best of luck to you.

 

[g.lemaitre]

Admittedly, I only found three questions--vector addition, matrix times its transpose, and computing projections--but it seems to me like for you, some aspects of the notation are not natural yet. For instance, the idea that a scalar times a vector multiplies all the elements of that vector. Well, maybe the idea makes sense on paper, but otherwise, the problems about vectors that you posted about have more to do with the arithmetic underlying the calculation than the concept itself.
Sorry I can't be of more help. Best of luck to you.

It took me a while to get my textbook digitized so that i could post questions about it. then unfortunately i relapsed into my chess addiction which took me away from my LA studies. i see chess as a horrible waste of time but to me at least it is profoundly addictive. i hadn't played in 7 years but playing just one game got me hooked on it again and i ended up wasting 20 hours of time spread over about 8 days. hopefully in the next few days i'll get back on it.

 

[Muphrid]

In lieu of scanning the text, some templates for column vectors and matrices in LaTeX might be of use.

A column vector:

<br /> \begin{bmatrix}<br /> 1 \\<br /> 3 \\<br /> -2<br /> \end{bmatrix}<br />

[tex]
\begin{bmatrix}
1 \\
3 \\
-2
\end{bmatrix}
[/tex]

A matrix:
<br /> \begin{bmatrix}<br /> 1 &amp; 3 &amp; 5 \\<br /> 2 &amp; -4 &amp; 7 \\<br /> -3 &amp; -6 &amp; 10<br /> \end{bmatrix}<br />

[tex]
\begin{bmatrix}
1 & 3 & 5 \\
2 & -4 & 7 \\
-3 & -6 & 10
\end{bmatrix}
[/tex]

A matrix equation:

<br /> \begin{bmatrix}<br /> 1 &amp; 3 &amp; 5 \\<br /> 2 &amp; -4 &amp; 7 \\<br /> -3 &amp; -6 &amp; 10<br /> \end{bmatrix}<br /> \begin{bmatrix}<br /> 1 \\<br /> 3 \\<br /> -2<br /> \end{bmatrix}<br /> =<br /> \begin{bmatrix}<br /> 0 \\<br /> -28 \\<br /> -41<br /> \end{bmatrix}<br />

[tex]
\begin{bmatrix}
1 & 3 & 5 \\
2 & -4 & 7 \\
-3 & -6 & 10
\end{bmatrix}
\begin{bmatrix}
1 \\
3 \\
-2
\end{bmatrix}
=
\begin{bmatrix}
0 \\
-28 \\
-41
\end{bmatrix}
[/tex]

Latex is totally insensitive to single line breaks (they're put in by hand with \\), so all that can be written more compactly once you're comfortable with it.

 

[20Tauri]

I thought LA was harder than calculus because many schools start abstract reasoning and proof-writing in LA, whereas a first course in calculus is mostly computational. In LA, you're likely to see somewhat intimidating topics like abstract vector spaces, whereas you wouldn't really see that level of abstraction in calculus unless you were taking advanced calc/real analysis.

 

[homeomorphic]

But not because LA is intrinsically more difficult than calc but because the pedagological tools for LA are seriously flawed. For calc you just put in the title of the problem you're working on in youtube and out pops about 10 videos that will walk you through how to do the problem. Plus the solution manual I had for calc was honestly easier to understand than the worked examples in the book, it had more steps.

There is a greater variety of difficulty levels in different linear algebra courses. Some are taught very abstractly, using vector spaces over arbitrary fields, and very heavy on the proofs. Some are based more on matrix calculations. There are also problems of greatly varying difficultly that profs can throw at you, depending on how difficult they want to make it.

There is a big difference between linear algebra and calculus. Calculus is for still aimed to be taught at fairly non-mathematical people. But once you get to linear algebra, it's assumed that most of the weaker students have been weeded out because probably most people who study linear algebra are there by choice, rather than because they are required to take it. It's for people who are reasonably comfortable with math. If you are a big math-phobic, it's not a good idea to study it or any subject that requires it. In particular, it's not for people who "need to see more steps" and always have example problems worked out for them.

The way it is taught may be pedagogically flawed, but it probably suffers more from pandering to people who are uncomfortable with being asked to understand concepts than it does from being made too difficult. By math majors standards, it's usually an easy class (to be fair I got a C in it, partly from not having figured out how to do math, partly from being late to more than one exam, and also, making dumb mistakes in some cases, even though I pretty much knew what I was doing). The more abstract courses can sometimes be putting the cart before the horse, asking students to accept too many (to them) unmotivated concepts. I would consider a course using arbitrary fields highly inappropriate for students just coming out of standard calculus classes. I never had a problem with problems being too hard or not understanding how to do things. The "how to do stuff" part isn't really that hard. What I did have a problem with is when people just focus on calculations in place of concepts. For example, introducing determinants, but never telling the students that determinants measure the volume of a parallelepiped spanned by the column vectors of a matrix.

For linear Algebra you don't have that. There are only 2 lecture series on LA by Strang and Khan and they aren't all that good. I'm on my 3rd LA text and they've all been pretty bad. The mistake they make is they assume the real basic of LA are too easy and they jump immediately to these abstruse problems that no one cares about. They're almost like brain teasers. I can't even master the mechanics of LA or the notation and then they shove these hard problems in my face without teaching me how to do the real easy stuff.

Without knowing what these "abstruse problems" are, it's hard to say. Maybe the abstruse problems are actually important to understanding the subject and knowing what's going on behind the scenes. It's also not clear what these basics are that are supposedly "not too easy". What if they are too easy for other students?

You don't want to lose the whole class, but on the other hand, you don't want to hold back the better students. Maybe they'll get bored of all that easy stuff.

 

[homeomorphic]

Anyone that does not like or work in pure mathematics has little need of proving mathematical theorms, only with being skillful at the tools of computing things and arithmetic.

I disagree because being able to prove theorems, and more importantly, being able to understand why things work, which is the key to proving theorems, actually helps with computing things. Particularly in linear algebra. Linear algebra would be incredibly dull if it were only matrix computations with no theory behind it. Of course, it's really more the intuition than proving the theorems that is useful.

You don't need to be a genius in linear algebra proofs to do quantum mechanics problems. To fully understand the theory, you might need to go deeper, but I find that its a waste of time to try and "intuitively" understand the matrix formulation since wave mechanics is soooo much easier to visualize, and matrix formulation is in my opinion strictly used for problem solving, especially for spin systems since wave mechanics isn't so nice with spin systems.

Waste of time to understand. Yep, that pretty much sums it up. Actually, the vector space approach to QM is more enlightening, I think, if you want to really understand quantum mechanics and feel like it's more of a natural thing that makes sense, rather than some mysterious think that someone else came up with for you to use. I think Dirac more or less demonstrated this from the start with his book. I'm a very visual thinker, by the way.

 

[Windowmaker]

I disagree with many of the posters here. Using resources online or solution manuals aren't doing the work for you. I used both and received over 100 percent in both calc 1 and 2. They are simply an aid and to ASSUME he's having them do his homework for his is extremely unjust because you don't know his circumstances.

As far as linear algebra goes, its not so bad in all reality. The hardest part was the proofs. Finding null spaces or col space and even eigenvalues/vectors wasn't extremely hard but to do a general proof to show something is linear without seeing it before is difficult. I got the lowest grade ever in a Math class in LA, which was a B+ which I blame on the proofs.

I also had the worse professor imaginable and literally half the class ended up failing. It's a difficult class. The khan videos weren't helpful. The way I passed was doing all the problems he assigned and then doing them each 2-3 times. And read the book. 95 percent of the class is understand the main concepts because the "math" is basically basic add or subtract or multiplication. If you don't know the what a kernel is in a transformation, you're not going to be able to do the proof or find it or if you don't know the definition of a basis how are you doing to prove it?

In calculus you didn't really need to understand what an integral was to solve it, it was just "backwards" derivatives. The concepts and vocab are just more important in LA that other math courses.

 

[homeomorphic]

I disagree with many of the posters here. Using resources online or solution manuals aren't doing the work for you. I used both and received over 100 percent in both calc 1 and 2. They are simply an aid and to ASSUME he's having them do his homework for his is extremely unjust because you don't know his circumstances.

It's not assuming that much, the way he worded things. If you are more independent-minded, you aren't going to be bothered that you can't find ten youtube videos showing you exactly how to do each problem, even providing more steps. Providing more steps. I mean, come on. Asking even for the little microscopic gaps to be filled. At some point, you have to start thinking for yourself a little more.

I think I have looked at about 3 worked solutions total in my life and I am finishing a PhD. How come I didn't need any solutions videos?

It's not that I'm too good to need them. I don't think anyone needs them because it's just not a good strategy. If something is too hard, you need hints, not a full solution with every last teeny-tiny step shown.

Some people receive great grades, but that doesn't mean that they actually learned anything. In high school, at one point I had a C in my trig class (in the end, I got a B-), but I was HELPING the people at the top of my class who had A's. I wasn't even that good at math back then. You can get good grades by cheating and parroting what you are told without even understanding it.

 

[Windowmaker]

But not because LA is intrinsically more difficult than calc but because the pedagological tools for LA are seriously flawed. For calc you just put in the title of the problem you're working on in youtube and out pops about 10 videos that will walk you through how to do the problem. Plus the solution manual I had for calc was honestly easier to understand than the worked examples in the book, it had more steps. For linear Algebra you don't have that. There are only 2 lecture series on LA by Strang and Khan and they aren't all that good. I'm on my 3rd LA text and they've all been pretty bad. The mistake they make is they assume the real basic of LA are too easy and they jump immediately to these abstruse problems that no one cares about. They're almost like brain teasers. I can't even master the mechanics of LA or the notation and then they shove these hard problems in my face without teaching me how to do the real easy stuff.

He's trying to understand the problems, looking at lectures for even more understanding and is using his third LA book. I think he's putting in much more effort than you are giving him credit for...

 

[g.lemaitre]

I decided to take a look at the second textbook by Hill, elementary linear algebra. It's much easier than the other two. I was able to get through two sections today having understood most of what was going on, so that's a relief.

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.