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A 2020 Vision of Linear Algebra

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Re: A 2020 Vision of Linear Algebra

#141

Post by basic_bgnr »

Phlogistique wrote: post #69
pengaru wrote: post #66

It's ridiculous how much random college-level linear algebra textbook material I stared at before things clicked in the course of just jumping in and exploring 3D graphics and writing my own 3D vector, matrix multiplication and 3D transform headers and using them in making some games in plain C. At some point it's like "Wait, is linear algebra really just about heaps of multiplication and addition? Like every dimensi…

Is it the same issue as the infamous "monads tutorials" problem, where the understanding takes a lot of time to infuse but looks obvious in retrospect when it finally clicks?

"whenever somebody gets a deeper understanding of monads, they immediately lose the ability to explain it to other" I don't remember where I've read this but it still holds even today.

Re: A 2020 Vision of Linear Algebra

#142

Post by vlasev »

jacobolus wrote: post #45
auggierose wrote: post #23

Have not watched the videos yet, but that seems to me more like an 1820 vision of linear algebra :-) If you look at the order of topics in his book "An Introduction to Linear Algebra", you will find the topic "Linear Transformation" way back in chapter 8! Even after the chapters eigenvalue decomposition and singular value decomposition. But understanding that a matrix is just the representation of a linear transforma…

Gaussian Elimination is indeed from the 1820s. All the rest is more recent than that. The idea of matrix decomposition per se comes from the 1850s. The earliest work on something like the SVD is from the 1870s. You are onto something though. Strang is coming from a direction of numerical computations and algorithms for solving real-world problems. Pure mathematics departments for at least the past maybe 80 years ofte…

I believe this is largely because in the field of mathematics, Linear algebra is just the seed that sprouts the growth of other very very useful mathematical subjects like Abstract Algebra, Functional Analysis and so on. Linear Algebra is used as a stepping stone to more general theories that are also super useful.

Take Hilbert spaces for example. They are based on linear algebra. They are quite general and you might argue that there's a lot of symbol twiddling there. However, Hilbert spaces are/were essential in the study of Quantum Mechanics, which we can argue is a very important topic.

And if you only stick with matrices and numerics, you're bound to get stuck in the numbers and details and miss the big picture. A lot of results are much cleaner to obtain once you divorce yourself from the concrete world of matrix representation.

Of course, we should probably have the best of both worlds. I'm not saying applications are unimportant. Take something like signal processing, which relies heavily on both numerics and general theory.

So I'd like to add something to your point. Math departments optimize the education of math students towards the more general, and perhaps students not interested in pursuing pure math should have course-work that reflects that.

Re: A 2020 Vision of Linear Algebra

#143

Post by Koshkin »

dleslie wrote: post #109
jerf wrote: post #107

Earlier quoted context omitted.

The problem is telling people it's a container is "over describing" it. We don't need to hypothesize about that. We have the space suits and burritos to prove it is not a good didactic approach. It is not removing from the definition to simplify, it is adding to the definition, exactly as I carefully showed in my description of "Iterator". An Iterator is "a thing that presents a series of items". It does not simplify…

Containers make sense. Abstract computer science doesn't. Part of why Haskell appears like such an implacable curmudgeon is the predilection of its community to believe that users must grasp type and logic theory to use it. They don't. Just like they don't need to have a mental model of their computer to write software for it.

In my experience, not having a mental model of the computer you are going to run your software on will bite you on the ass sooner or later.

Re: A 2020 Vision of Linear Algebra

#144

Post by howling »

Koshkin wrote: post #128
cbm-vic-20 wrote: post #96

Earlier quoted context omitted.

"A monad is just a monoid in the category of endofunctors, what's the problem?"

I don't think this definition is correct. (A monad is an endofunctor.)

The definition is correct. A monad is an endofunctor with return and join functions. Just like a monoid in the category of sets is a set with identity and multiplication.

Re: A 2020 Vision of Linear Algebra

#145

Post by vlasev »

threatofrain wrote: post #65
enitihas wrote: post #35

Another good Linear Algebra book is "Linear Algebra Done Right", which Springer is giving for free right now. Link: https://link.springer.com/book/10.1007/978-3-319-11080-6

IMO Axler's book should be read either during or after you take an introductory course on Linear Algebra. > You are probably about to begin your second exposure to linear algebra. Unlike your first brush with the subject, which probably emphasized Euclidean spaces and matrices, this encounter will focus on abstract vector spaces and linear maps.

I whole-heartedly agree. Axler's book is a great stepping stone to more abstract linear algebra.

Re: A 2020 Vision of Linear Algebra

#146

Post by Bootvis »

heinrichhartman wrote: post #139
Bootvis wrote: post #131

Earlier quoted context omitted.

I don’t have an intuition for these concepts I’m afraid (I probably should watch the videos). What I don’t see for instance is how this relates to the fact that a matrix A with det(A) = 0 is not invertible.

The geometric version is the most intuitive for me: If the volume of the prallel-epiped is zero, then there will be directions in the target space, that you did not hit. Hence he matrix can not be invertible.

I will have to understand what directions in target space are first. I’ll guess I’ll have to do the work ;)

Re: A 2020 Vision of Linear Algebra

#147

Post by justinmeiners »

anandrm wrote: post #84

Just curious .. what really are the usecases where of Linear Algebra is applied ? Any domain of software development ?

- scale or rotate an image.

- root finding algorithm with more than one variable.

- graph problems like Google's PageRank

- statistical analysis

- 3d rendering (projecting a 3d scene onto a 2d image)

- solving systems of equation (also see linear programming)

Linear algebra is very basic and fundamental to physics and math.

Re: A 2020 Vision of Linear Algebra

#148

Post by vlasev »

Bootvis wrote: post #131
heinrichhartman wrote: post #27

Earlier quoted context omitted.

> seldom get back the "best" conceptual answer, "the determinant is the product of the eigenvalues." I found that the "best conceptual" answer depends a lot on taste, and what concepts you are familiar with. In this case: - Calculating exact eigenvalues of matrices larger than 4x4 is impractical, since it requires you to solve a polynomial of degree >4. - The EV exist only in algebraically closed fields (complex numb…

I don’t have an intuition for these concepts I’m afraid (I probably should watch the videos). What I don’t see for instance is how this relates to the fact that a matrix A with det(A) = 0 is not invertible.

Take a 3x3 matrix A for example. Then det(A) is the volume of the parallepiped formed by the row vectors. If one vector is a linear combination of the other two, this means that the vectors lie in a plane, which has volume 0 in 3D, so det(A) = 0. Since we have a plane in 3D, this means A can't express all vectors in 3D, so it's not invertible. This generalizes to any dimension.

Re: A 2020 Vision of Linear Algebra

#149

Post by vajrabum »

DagAgren wrote: post #79
Phlogistique wrote: post #69

Earlier quoted context omitted.

Is it the same issue as the infamous "monads tutorials" problem, where the understanding takes a lot of time to infuse but looks obvious in retrospect when it finally clicks?

I am convinced that monads induce a very specific kind of brain damage that makes a person incapable of ever explaining monads.

This explanation did it for me. https://www.reddit.com/r/math/comments/ap25mr/a_monad_is_a_m...

Re: A 2020 Vision of Linear Algebra

#150

Post by Koshkin »

auggierose wrote: post #23

Have not watched the videos yet, but that seems to me more like an 1820 vision of linear algebra :-) If you look at the order of topics in his book "An Introduction to Linear Algebra", you will find the topic "Linear Transformation" way back in chapter 8! Even after the chapters eigenvalue decomposition and singular value decomposition. But understanding that a matrix is just the representation of a linear transforma…

> a matrix is just the representation of a linear transformation

While this view certainly helps intuition at initial stages of learning, it is not "just" that, and computational methods involving matrices are of much more practical importance (similar to being able to add and multiply numbers which we are taught early in life) which is probably why the stress is on them first and foremost. Someone said, "learn to calculate, understanding will come later."

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