Current Events > Am I the only one who finds linear algebra harder than integral calculus?

(+) Yeah! (+)

I did integral calculus last semester and I did very well. I had some trouble grasping stuff like Taylor and Mclaurin series at the end but with some practice I was ok

But linear algebra is a whole other ball game. I just find it ten times harder.

Like... I can easily memorize all the rules and theory. But then I have to do row operations and it's like... over and over I have stuff like -5/4 minus -17/82 times -41/113 and things like that. I always end up punching the numbers wrong on the calculator and then I end up with matrices that have entries like 218/1473 whereas the solution has a nice clean matrix with just integers or simple fractions

Why is it so hard?
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For me, linear algebra is harder just because I don't have as much experience in it as I do in calculus

I didn't have linear algebra until Uni, whereas we were already solving separable ODEs at the end of high school
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row operations are freaking easy... its the rest of the stuff that makes no sense to me
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your calculator can also check solve matrices for u btw so u can see what the final solution is supposed to look like
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Nope.

Linear algebra was one of the hardest math classes I took in college.

Of course that could be because I never went to class or did homework. Remind me to tell you that story sometime.

But also when I taught myself half the class material for each of the exams, those were two really confusing cram sessions.
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I've used Eigenvalues and Eigenvectors extensively in my Quantum Mechanics courses, Electromagnetism courses, and in a Computer Animation course a while back. I still don't know what an Eigenvalue or Eigenvector is.
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Linear Algebra at most places requires integral calculus...so it's unsurprising that it's harder.
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Idk I stopped taking math classes like 10 years ago because I'm not a scrub.
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Sad_Face posted...

I've used Eigenvalues and Eigenvectors extensively in my Quantum Mechanics courses, Electromagnetism courses, and in a Computer Animation course a while back. I still don't know what an Eigenvalue or Eigenvector is.

It's something to do with matrices.

And like, multiplying them. Somehow if you multiply an eigenvector by a matrix (or was it the other way around) you get... something.
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An eigenvector of a matrix is a vector where applying the matrix is the same as multiplying the vector by a scalar (which is the eigenvalue).
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DevsBro posted...

Sad_Face posted...

I've used Eigenvalues and Eigenvectors extensively in my Quantum Mechanics courses, Electromagnetism courses, and in a Computer Animation course a while back. I still don't know what an Eigenvalue or Eigenvector is.

It's something to do with matrices.

And like, multiplying them. Somehow if you multiply an eigenvector by a matrix (or was it the other way around) you get... something.

Oh, I know how to solve for Eigenvectors. Computer Animation, you can find use them to find non trivial solutions to the number of collisions on a rigid bodies at a point and time. In E&M... it's been so long, I can't remember. But what Eigenvectors/values mean abstractly, I have no clue. And the context changes with each field it's applied to. At least with a derivative, you know it's a rate of change, how quickly something changes with respect to an element. I don't know how to summarize EVs like that.
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I had to review eigenvectors and eigenvalues a few months ago. I've also taken 3 linear algebra courses and a few related courses >_>

So I'll probably be in the same boat this time next year reviewing crap if I don't use it in my current project. Shit is confusing if you don't use it or just have Matlab, R, or Python do everything for you

If you guys want a refresher and don't feel like reading, I recommend the 3blue1brown essence of linear algebra playlist on youtube
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MacadamianNut3 posted...

I had to review eigenvectors and eigenvalues a few months ago. I've also taken 3 linear algebra courses and a few related courses >_>

So I'll probably be in the same boat this time next year reviewing crap if I don't use it in my current project. Shit is confusing if you don't use it or just have Matlab, R, or Python do everything for you

If you guys want a refresher and don't feel like reading, I recommend the 3blue1brown essence of linear algebra playlist on youtube

Where was this when I was taking maths 2 last year, huh????
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Sad_Face posted...

Oh, I know how to solve for Eigenvectors. Computer Animation, you can find use them to find non trivial solutions to the number of collisions on a rigid bodies at a point and time. In E&M... it's been so long, I can't remember. But what Eigenvectors/values mean abstractly, I have no clue. And the context changes with each field it's applied to. At least with a derivative, you know it's a rate of change, how quickly something changes with respect to an element. I don't know how to summarize EVs like that.

So you understand the significance of fixed points, right?
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I feel ya Sad Face. I knew how to calculate determinants no problem for years, but I didn't actually know that it just described how much an area is scaled after applying the matrix transformation until 4 months ago

None of my math teachers took the time to flat out state this

teepan95 posted...

maths

trigger warning
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Matrix math just didn't make sense to me, I could never remember the pattern of how it worked, so linear algebra was hard. I could do fine if I could use a calculator for the matrix math part.
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MacadamianNut3 posted...

I feel ya Sad Face. I knew how to calculate determinants no problem for years, but I didn't actually know that it just described how much an area is scaled after applying the matrix transformation until 4 months ago

None of my math teachers took the time to flat out state this

teepan95 posted...

maths

trigger warning

Wow your mathS teachers sucked
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Wait until you learn measure theory

Anteaterking posted...

So you understand the significance of fixed points, right?

Absolutely not. I have no clue what you're talking about. Please, edumacate me.

Honestly this is the reason why I bombed my QM courses way back when. It was just calculus and some linear algebra. But the thing is the homeworks were tedious, so I got fed up doing my homework every single week (I have better things to do than to solve integrals with a billion and 2 variables for pages upon pages, like other homework and sleep), so I gave up halfway through not realizing that the actual theory that makes the HW worth it comes at the very end of the problems. When exam time comes out, that's when I learned the subject. And deeply regretted not finishing my HW problems.

MacadamianNut3 posted...

feel ya Sad Face. I knew how to calculate determinants no problem for years, but I didn't actually know that it just described how much an area is scaled after applying the matrix transformation until 4 months ago

None of my math teachers took the time to flat out state this

Really? Okay, that makes sense. I can see it now if you look at the Magnetic Force's equation, qv X B, (charge, velocity of a particle, Magnetic Field strength) and you can take the determinant to get the magnitude of the Force, or area of the 2 vectors.
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You know what really blew my mind?

When I was reading up on quaternions and decided to multiply two of them together using FOIL (or whatever the four-term equivalent is), I literally ended up with the difference of the dot product and the cross product of two vectors. Suddenly those formulas got a lot less arbitrary.

I really do wonder sometimes why we teach math backwards. Sure, sometimes memorizing a fromula is easier for kids than teaching them the calculus used to derive it, but stuff like that? It's a million times easier to remember the cross product formula knowing it's basicall just FOIL on a quaternion. Quaternions aren't even a complicated concept... well except for noncommutative multiplication.
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Realizing what the determinant is made matrix inverses obvious to me and why a matrix with a 0 determinant doesn't have an inverse (there's no transformation you can use to take an area squashed down to zero and blow it back up to its original size. You lost at least 1 dimension)

If only it was explained like this to me 10 years ago
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Sad_Face posted...

Absolutely not. I have no clue what you're talking about. Please, edumacate me.

So it's sometimes interesting to consider the things that are fixed by a transformation. For example, if I have a transformation f(x)=4x, the point x=0 is the only fixed point (since f(0)=0). In higher dimensions, like with vector spaces it would be like saying that a linear transformation defined by a matrix fixes a certain point (or the vector from the origin to that point).

Eigenvectors are an extension of that, where instead of saying "I want to know what points (vectors) are fixed by this transformation" you instead say "I want to know which vectors just stretch or contract, but don't change where they are pointing".

So say for some matrix M, [1,2] is an eigenvector with eigenvalue 2. That means when you apply your transformation M to the vector [1,2], it becomes [2,4]. It was the equivalent of just multiplying each coordinate by 2, so it's pointing in the same direction , just longer.
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Initially I found linear algebra much harder. But between four different courses using it extensively (regular linear, applied linear, numerical methods, intermediate differential equations) I've gotten more used to it so it's more even now.
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Linear Algebra, as a course, was more difficult for me than my integral calculus course was.

One reason for this was because it felt like all the algebraic intuition I had built up over the years was useless. It was still useful in calc 2, but in linear algebra it's like you're working with this new realm of matrices and the algebraic intuition doesn't help much. Particularly for the last part of my course, I basically had no intuition whatsoever about what was going on. I just basically had to memorize a series of steps about matrix operations and then go through that memorized process without actually having any intuition about what I was doing.

Also my linear algebra professor had all these weekly pop quizzes and it was tough to know what to study for because sometimes it would be based on the homework that was due that week, while other times it was based on the content taught the previous day, which would be like the section ahead of that week's homework -.-
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MacadamianNut3 posted...

Realizing what the determinant is made matrix inverses obvious to me and why a matrix with a 0 determinant doesn't have an inverse (there's no transformation you can use to take an area squashed down to zero and blow it back up to its original size. You lost at least 1 dimension)

If only it was explained like this to me 10 years ago

I made this image during my linear algebra class to help me remember this kind of stuff >_>



The weird al pic is cuz our professor used this pic as some linear transformation example thing
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Yea intuition helps a bunch in linear.

I recommend 3Blue1Brown's series on the topic on YouTube. It goes over some fundamentals of it in a lot more palatable way.

He also has a calculus series fwiw
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Personally I find linear algebra a lot easier. That probably comes from the fact that I've applied it extensively during my uni studies, since I took game engineering. Not a lot of calculus there. Figuring out tricky limits or substitutions was always harder than the kinds of linear algebra problems I solved.

Eigenvectors/values can be nifty for a lot of thing, but the principle is quite simple. Keep in mind that "eigen" is German and means "own". An eigenvector of a transformation will be mapped onto its own line. Consider projection onto a plane. Any vector that is already in the plane will be projected on itself. So those are the eigenvectors of that transformation.
I think it's by far easiest to grasp this if you take some transformations in R or R that are easy to visualize and understand, then examine those eigenvectors. What are the eigenvectors of a rotation? What are the eigenvectors or a reflection? Etc.
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yes
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when i was at University my calculus classes were numbered in the 150s while the Linear Algebra was like 540-something.
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DevsBro posted...

I really do wonder sometimes why we teach math backwards. Sure, sometimes memorizing a fromula is easier for kids than teaching them the calculus used to derive it, but stuff like that? It's a million times easier to remember the cross product formula knowing it's basicall just FOIL on a quaternion. Quaternions aren't even a complicated concept... well except for noncommutative multiplication.

What I dislike about how math is taught, is how we learn the tools before the problems when tools are created to solve problems that exist. It's more intuitive to learn the problem first and then steps we take to solve them.

Anteaterking posted...

Eigenvectors are an extension of that, where instead of saying "I want to know what points (vectors) are fixed by this transformation" you instead say "I want to know which vectors just stretch or contract, but don't change where they are pointing".

So say for some matrix M, [1,2] is an eigenvector with eigenvalue 2. That means when you apply your transformation M to the vector [1,2], it becomes [2,4]. It was the equivalent of just multiplying each coordinate by 2, so it's pointing in the same direction , just longer.

scar the 1 posted...

Eigenvectors/values can be nifty for a lot of thing, but the principle is quite simple. Keep in mind that "eigen" is German and means "own". An eigenvector of a transformation will be mapped onto its own line. Consider projection onto a plane. Any vector that is already in the plane will be projected on itself. So those are the eigenvectors of that transformation.
I think it's by far easiest to grasp this if you take some transformations in R or R that are easy to visualize and understand, then examine those eigenvectors. What are the eigenvectors of a rotation? What are the eigenvectors or a reflection? Etc.

These make total sense. Thank you.
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The amount of memorization required for linear algebra is a lot higher than integral calculus.

You could derive most of the calculus formulas yourself when you got stuck during test, but if you forget even just 1 important property of vectors, that question is gone.

TrumpTrain posted...

The amount of memorization required for linear algebra is a lot higher than integral calculus.

You could derive most of the calculus formulas yourself when you got stuck during test, but if you forget even just 1 important property of vectors, that question is gone.

I completely disagree. In fact I think that most things you memorize in linear algebra are quite easy to show. However in calc it's a nightmare to show e.g., the substitution needed for indefinite integral of dx/sqrt(a+x), so you're better off just remembering that shit.
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