Trying to review linear algebra after graduating

[CookieSalesman]

TL;DR Summary: linear algebra confusing & hard; how to study right?

I plan to return to linear because that was one of the classes that you couldn't just try to shamble your way through somewhat. multivar calculus theorems still make some sense visually. Linear algebra exams were like trying to imagine the 5th dimension and I felt like the homework wasn't even remotely similar to exam content and had no idea how to prepare. I would look at the homework and go 'yeah that makes sense, I can probably get a B+' and then the exam question 1 would be written in hieroglyphics. This did wonders for my morale.

I'm trying to go over the basics, starting from calculus 1 (eventually will reach linear alg) But I'm at the dot product and cross product and this stuff makes no sense already. This time I want to understand everything deeply and have no questions.

Well, I understand the justification for calculus itself but not the dot product. Why the heck are you turning vectors into numbers? It seems like an arbitrary operation. And the cross product is just that but worse. The cross product has equivalences, too, where you can calculated the cross different ways. I mean, I've got my own equation too! We just chop off the right column of each matrix. How about that? Why isn't that operation mentioned in the textbook?

This is deeply weird, because this is basically just reminding me of high school spray and pray physics where you just threw numbers into equations as long you have one number for each variable right? - and just memorized the equations. How is this d.p and cxp in the textbook different at all?? You just gave me two operations (presumably written on the stone tablet handed to Moses) and just instructed me to use them. Right? Arguably, it's the exact same stuff, isn't it?

But what's frustrating is that the textbook (rogowski, adams, franz) doesn't explain where this stuff is coming from. Here I am trying to figure this stuff out seriously and these equations for d.p and cxp just spontaneously appear in the world? What?
~~~


Well, the above problem is actually not the focus - it's just an example of the problem.

I feel like I don't want to revisit math in the convoluted collegiate context again.

Let me give two examples: It would be less frustrating if I understood where things like the dot product came from. Otherwise, everything is just algebraic manipulation of equations that chose to spontaneously appear in the world. Why does the dot product even have the right to exist? Well, the textbook sure doesn't know!

And furthermore second example, I only recently asked myself, how the hell is it that I can just take the log of both sides of an equation? Squaring an equation - makes sense to me. But applying a logarithm? I can't really explain it but that feels like a wholly different operation. But yet we just 'do it'. No one explains this! It's not even an educator problem, the textbook won't even tell you!

Am I just reading the wrong textbooks? How to study this convoluted and cancerous discipline that presumes to haunt my existence?

[Mentor Note: Post has been edited to remove extensive profanity]

 

[martinbn]

The whole post seems to me that you just want to vent and you are not really interested in any answers, but this I find surprising

And furthermore, I only recently asked myself, how the hell is it that I can just take the log of both sides of an equation? Squaring an equation - makes sense to me. But applying a logarithm?

You have an equality $A=B$, which means that the two quantities $A$ and $B$ are equal. Then you take log of them (remember they are the same) and you get $log(A)=log(B)$, and you ask why!!! What else? If you take log of one and the same number you get the same result! Isn't that obvious to you!

 

[CookieSalesman]

^ nope, I didn't think of it that way, and it's probably because last time I asked this someone said something about monotonic functions.

I am slightly frustrated that this focuses on an example - an example I use to support some points - and does not focus on any of the points, while I appreciate the explanation itself, but yes, I do have an actual serious question and I am incredibly and unambiguously serious about asking it.

 

[PeroK]

First, although it's politically incorrect to say so, some of us do naturally understand mathematics and come up with ideas ourselves without having to be taught them. Other people are like that with music, drawing, writing or sport etc. For me, there was never a question of memorising things or just putting numbers into equations. It all seemed to make sense to me.

If mathematics makes little sense to you, then I sympathise, but you might as well complain that you don't understand music or poetry or can't throw a ball.

If you are learning from textbooks, then it's less likely that the author will have time to answers everyone's questions in the text. Perhaps this forum can fill the gaps - I guess that's why you joined.

However, you will have to adopt a positive attitude. You'll get further and people will be more willing to help you if you accept that mathematics is what it is and it's you that has to adapt to become competent at it.

Finally, enlightenment doesn't necessarily come all at once. You may have to accept certain things to begin with and then in time hope that it becomes clear.

 

[BvU]

I am slightly frustrated that this focuses on an example - an example I use to support some points - and does not focus on any of the points, while I appreciate the explanation itself, but yes, I do have an actual serious question and I am incredibly and unambiguously serious about asking it.

Seems to me you started with examples...

I am surprised you wonder about (for example ) dot and cross products and at the same time about Planck lengths and the looks of black holes. Were you as critical as this in kindergarten and primary school ? But still learned to read and write and do some elementary math ?

Can you not imagine that all the worlds university math/physics/science curricula contain relevant stuff to lay a foundation for more advanced stages ?

I recognize some of your experience, for example () in first year at university (1971) the lectures were so good that most of us students thought tests and exams would be a breeze. They were not. Not because they were in hieroglyphs. (what are hieroglyphics ?)

What worked wonders for me was to actively engage in the workshops (= doing excercises. NOT: watching others solve them). Another thing that helped: looking at things from the other perspective: how can one compose meaningful exercises and exam problems from the material taught. That's actually quite a formidable task (and I grant you that a lot of mediocre universities fail miserably at that -- even without falling back on hieroglyphs)

I don't think you are reading the wrong textbooks. You are just reading them wrong. They are meant to teach, not intended or meant to justify what they teach.

And I never wondered about 5d.

$\$

 

[CookieSalesman]

But still learned to read and write and do some elementary math ?

This is true, but I am sitting here wondering what the dot and cross product are all about and the textbook doesn't say anything.

Thank you for mentioning workshops or the 'other' perspective. I was trying to start doing that in the math actually.

Can you elaborate what is the workshop? Is that just the sessions that are paired with classes where you do homework?

It's good to get your thoughts, thanks.

 

[PeroK]

This is true, but I am sitting here wondering what the dot and cross product are all about and the textbook doesn't say anything.

You have the Internet at your disposal. I tried "scalar product history" and got this first up:

https://math.stackexchange.com/questions/62318/origin-of-the-dot-and-cross-product

There are many examples of the scalar product in physics and applied mathematics. And also in studying 2D and 3D geometry. Here's the equation of a plane using the scalar product extensively:

https://tutorial.math.lamar.edu/classes/calciii/eqnsofplanes.aspx

PS Paul's online notes is a great reference for all things calculus.

The scalar product generalises to abstract vector spaces (it's usually called an inner product by then) and the formalism of Quantum Mechanics is based on vectors, operators and the inner product:

https://en.wikipedia.org/wiki/Bra–ket_notation

General and Special Relativity uses four-vectors (these are 3D spatial vectors with an added four component in the time dimension) and again the inner product is central to the development of the theory.

The dot/scalar/inner product is ubiquitous in pure and applied mathematics and physics.

The question for any student these days (with the Internet) is how far to burrow down this rabbit hole and how much to focus on learning the material at hand. You could spend the next week or month or year investigating the inner product in all its forms. Or, you could accept that the inner product (and to some extent the cross product) are two of the most useful things you'll ever learn and get on with it.

 

[BvU]

Can you elaborate what is the workshop? Is that just the sessions that are paired with classes where you do homework?

Yes. In dutch they are called 'werkcollege': you work on exercises and senior students are paid to circulate and help out. Like in PF, they don't give the answers, but ask guiding questions and sometimes give hints. It's actually also a pretty good training for social skills

Sometimes there is a bonus, e.g. when towards the exam the professor also shows up and drops small hints about which exercises would make good exam questions .

$\$

 

[CookieSalesman]

Thanks to both.

 

[MidgetDwarf]

If you want to truly understand these mathematical concepts, then you have to be read a pure math book in these subjects. [Not easy]

Or you can approach it from an engineering/physics perspective and gain an intuitive understanding.

 

[WWGD]

There's something to be said for the " fake it until you make it" approach. You do your best to find ways of solving your problem, then, one day some time afterwards, it has just sunk in and makes sense. You can't really always just push yourself to make sense of it then and there , at the moment. But the wheels keep spinning in the back of your mind.

 

[PeroK]

There's something to be said for the " fake it until you make it" approach.

Is that your strategy?

 

[MidgetDwarf]

TL;DR Summary: linear algebra confusing & hard; how to study right?

I plan to return to linear because that was one of the classes that you couldn't just try to shamble your way through somewhat. multivar calculus theorems still make some sense visually. Linear algebra exams were like trying to imagine the 5th dimension and I felt like the homework wasn't even remotely similar to exam content and had no idea how to prepare. I would look at the homework and go 'yeah that makes sense, I can probably get a B+' and then the exam question 1 would be written in hieroglyphics. This did wonders for my morale.

I'm trying to go over the basics, starting from calculus 1 (eventually will reach linear alg) But I'm at the dot product and cross product and this stuff makes no sense already. This time I want to understand everything deeply and have no questions.

Well, I understand the justification for calculus itself but not the dot product. Why the heck are you turning vectors into numbers? It seems like an arbitrary operation. And the cross product is just that but worse. The cross product has equivalences, too, where you can calculated the cross different ways. I mean, I've got my own equation too! We just chop off the right column of each matrix. How about that? Why isn't that operation mentioned in the textbook?

This is deeply weird, because this is basically just reminding me of high school spray and pray physics where you just threw numbers into equations as long you have one number for each variable right? - and just memorized the equations. How is this d.p and cxp in the textbook different at all?? You just gave me two operations (presumably written on the stone tablet handed to Moses) and just instructed me to use them. Right? Arguably, it's the exact same stuff, isn't it?

But what's frustrating is that the textbook (rogowski, adams, franz) doesn't explain where this stuff is coming from. Here I am trying to figure this stuff out seriously and these equations for d.p and cxp just spontaneously appear in the world? What?
~~~


Well, the above problem is actually not the focus - it's just an example of the problem.

I feel like I don't want to revisit math in the convoluted collegiate context again.

Let me give two examples: It would be less frustrating if I understood where things like the dot product came from. Otherwise, everything is just algebraic manipulation of equations that chose to spontaneously appear in the world. Why does the dot product even have the right to exist? Well, the textbook sure doesn't know!

And furthermore second example, I only recently asked myself, how the hell is it that I can just take the log of both sides of an equation? Squaring an equation - makes sense to me. But applying a logarithm? I can't really explain it but that feels like a wholly different operation. But yet we just 'do it'. No one explains this! It's not even an educator problem, the textbook won't even tell you!

Am I just reading the wrong textbooks? How to study this convoluted and cancerous discipline that presumes to haunt my existence?

[Mentor Note: Post has been edited to remove extensive profanity]

Hmm upon looking at my book collection.

I think Paul Shields, short well written but elementary, book on LA may be of use. In understanding Linear Algebra, or at least intuitively.

If you want something grown up, but still an engineering book which will only give you an intuitive explanation, then look at the book by Anton:Elementary Linear Algebra.

The notions you are struggling with made more sense to me as I read Linear Algebra Done Right (a pure mathbook), and studying this material again while learning Field Theory (again pretty advanced stuff).

For the log part, I first learned why this works from Edwin E. Moise: Calculus. A book which has not been in print for a long time. Copies were $20, but are now upwards of $80 (if you can find it). It actually uses calculus to define the exponential functions and e itself, then uses that to explain what logs are.

I do not recall if Apostol or Courant does such a thing in there calculus text.

If you want to see how LA applies to calculus, then the book by Hubbard and Hubbard; Vector Calculus, Linear Algebra, and Differential Forms, is a good choice. Just download the errata from them.

It also gives an explanation of the scalar and cross product. A bit more computational, but shows its application (IRC). I have read too many books, and sometimes I loose track of what specifics are in what, unless its something aimed at senior in mathematics.



What did you graduate in?

 

[WWGD]

Is that your strategy?

Yes, it's worked at times. Why not?

 

[Vanadium 50]

I thought I had replied to this, but it must have been a similar thread. I used Anton, and its OK for what it tried to be.

There are basically two linear algebras out there. One is way of solving systems of equations, and the other is sort of an abstract algebra-lite. The first question is "which one do you want?"

Mismatches are not good. When I took it, the prof wanted to teach the 2nd one, and we wanted to learn the 1st. I would say Anton was more on our side than his.

FWIW. as a non-mathematician, I am OK with the more applied direction. Would I be a better person if I learned more? Maybe. But at some point one must leave the classroom, leave some things uncovered, and go out and do things.

 

[mpresic3]

I am having a hard time from your post finding out where your problem lies. On the one hand, you seem to think you think you can do homework problems and get about 85% on a test, indicating you think you have some mastery of the material. Then you mention the exam comes and you do not even understand the question?

I can relate to this with some experience, I have taking some liberty and conforming it to linear algebra.

I read HW problems and they are like: take the dot product of vectors (1, 2, 4), and (4,3,1). or take the cross product of those two vectors. Then the test comes and it says, prove the steinitz principle in linear algebra. Complete a vector which constitutes a basis along with the two vectors given.

One form of question asks for a very "mechanical" calculation, and the other question asks a theoretical question or a proof. In my school, engineers took a linear algebra with calculations which could be very involved but quite straightforward. In the honors linear algebra class, (after a horrific first exam), my instructor told me he would never ask an engineering type question. He was trying to probe how well we could understand the theory. The exams always looked cryptic.

You may be misplaced. Do you want to learn the theory or the applications. It could be quite a struggle to learn both.

On the other hand, you seem to recoil at learning even the "mechanical" parts of linear algebra, like dot and cross products. It sounds like you want to know what it is good for. Most math texts treat this lightly and rely on physics and engineering classes to show this.

In one real life example:
My graduate school roommate was a ham radio operator. He wanted me to tell him if I could find an equation that found the distance between two locations in two latitiude and two longitudes. (This was in the days before he could google it)

I first found the three vector from the center of Earth to the point with latitude and longitude 1.
Next,I found the three vector from the center of Earth to the point with latitude and longitude 2.
Then I found the dot product between these two vectors.
Then I used the fact that the angle between these vectors was the arc cosine of the dot product.
The distance between these points on the sphere is the radius times the arc cosine.

So you see, there are reasons to learn this.

In my job (something like surveying), I use the dot and cross products most everyday. My advice is learn the mechanics without complaining and have faith it will be useful in the future, or ask someone in this forum for an application.

 

[Muu9]

This is a bit late, but do you think this would have helped if you had seen it when you were having trouble understanding the cross product? https://betterexplained.com/articles/cross-product/