Relearning Matrices as Linear Functions • Hacker News Board

Re: Relearning Matrices as Linear Functions

#41

Post by rocqua » Tue, Apr 23, 2019, 9:34 PM UTC

throwawaymath wrote: post #36

_v7gu wrote: post #23

Earlier quoted context omitted.

Axler's book has the advantage of skipping determinants in order to provide a more intuitive approach to linear algebra.

I strongly disagree skipping determinants provides a more intuitive approach to linear algebra. I don't know your background, but I'd venture a guess you feel it does because the Laplace expansion formula for computing the determinant[1] feels uninspired and out of place. The reason determinants are hard to teach (in my opinion) is because a rigorous derivation of their formula isn't possible without first teaching m…

I agree, the way I still see determinant is as the 'volume scaling factor' of a linear transformation.

This means it makes sense that det(A) = 0 means A is non-invertible. It also makes a lot of sense when the jacobian pops up in the multi-dimensional chain rule.

Given the above, and the Cayley–Hamilton theorem, I never really had to know why the determinant was calculated the way it is. The above give enough of an interface to work with it.

Re: Relearning Matrices as Linear Functions

#42

Post by rocqua » Tue, Apr 23, 2019, 9:37 PM UTC

wolfgke wrote: post #28

mikorym wrote: post #24

The next relearning step is to construct the category where arrows are matrices...

> The next relearning step is to construct the category where arrows are matrices... Why not the category of vector spaces (morphisms are linear maps)?

Isn't that the same?

I suppose that technically, the 'arrows are matrices' definition rules out infinite dimensional vector spaces, but I'd guess that OP meant to include them.

An argument against would be to keep to a small category.

Re: Relearning Matrices as Linear Functions

#43

Post by throwawaymath » Tue, Apr 23, 2019, 9:42 PM UTC

wolfgke wrote: post #26

avip wrote: post #20

Earlier quoted context omitted.

>geometrically, all linear maps can be thought of as rotations and scalings. and reflections.

Shearings cannot be represented in this way.

Yes they can. This follows from singular value decomposition. Let S be the matrix representation of a shear transformation. There exist rotation matrices R, B and a diagonal matrix D such that S = RDC, where C is the transpose of B. D is the matrix representation of a scaling transformation and R, B are the matrix representations of rotation transformations. Since S is a product of rotation and scaling matrices, its corresponding linear transformation is a composition of rotations and scalings.

It would ordinarily be weird to represent shear transformations using rotations and scalings because shear matrices are elementary. But it checks out.

Re: Relearning Matrices as Linear Functions

#44

Post by rocqua » Tue, Apr 23, 2019, 9:43 PM UTC

kregasaurusrex wrote: post #12

Having not taken a linear algebra course in college, does anyone have a recommendation for a book/course to follow?

That would heavily depend on whether you are coming at it from a theoretical math p.o.v. or a more applied p.o.v.

Not that the applied approach should leave out the theory, because theoretical stuff like this article give a great and intuitive understanding of linear algebra. However, the more theoretical treatments should set up things like rings, modules, and even category theory that are much less useful from an applied perspective.

For the theoretical approach I've heard good things about 'linear algebra done right'. I imagine it is less appealing for the applied approach. All I can say is be wary of the 'shut up and calculate' mindset in linear algebra. Getting the ideas behind the concepts is essentially a shortcut to understanding linear algebra without any downsides.

Re: Relearning Matrices as Linear Functions

#45

Post by jefft255 » Tue, Apr 23, 2019, 9:48 PM UTC

brianberns wrote: post #32

msla wrote: post #19

Earlier quoted context omitted.

What got me for a while was the concept of a tensor: For example: What is a tensor? Wrong way to answer it: Well, the number 5 is a tensor. So's a row vector. So's a column vector. So's the dot product and the cross product. So's a two-dimensional matrix. So's a four-dimensional matrix, just... don't ask me to write one on the board, eh? So's this Greek letter with smaller Greek letters arranged on its top right and…

As a layman, the word "tensor" always intimidated me. As a programmer, I was surprised then when I found out that a tensor is just a multi-dimensional array (where the number of dimensions can be as small as 0). That was a concept I was already quite comfortable with.

That’s not really what a tensor is; this simplification is due to tensorflow I think?

Re: Relearning Matrices as Linear Functions

#46

Post by adamnemecek » Tue, Apr 23, 2019, 9:53 PM UTC

Conjugate transpose and other adjoints are kinda nuts, they are the other part of the story

http://www.reproducibility.org/RSF/book/bei/conj/paper_html/...

Esp the ray tracing/topology relationship is nuts.

Re: Relearning Matrices as Linear Functions

#47

Post by Retra » Tue, Apr 23, 2019, 10:05 PM UTC

brianberns wrote: post #32

msla wrote: post #19

Earlier quoted context omitted.

What got me for a while was the concept of a tensor: For example: What is a tensor? Wrong way to answer it: Well, the number 5 is a tensor. So's a row vector. So's a column vector. So's the dot product and the cross product. So's a two-dimensional matrix. So's a four-dimensional matrix, just... don't ask me to write one on the board, eh? So's this Greek letter with smaller Greek letters arranged on its top right and…

As a layman, the word "tensor" always intimidated me. As a programmer, I was surprised then when I found out that a tensor is just a multi-dimensional array (where the number of dimensions can be as small as 0). That was a concept I was already quite comfortable with.

You're confusing a tensor with its representation. Tensors are objects which must obey a certain set of rules. (Which rules depends on whether you're talking to a mathematician or a physicist.)

Re: Relearning Matrices as Linear Functions

#48

Post by postsantum » Tue, Apr 23, 2019, 10:09 PM UTC

_hardwaregeek wrote: post #31

Linear Algebra, at least at my school, is taught pretty poorly. Instead of teaching the beauty of transformations, the course is boggled down in numerical nonsense and tedious calculations (who wants to find the inverse of a 3x3 matrix? Bueller? Bueller?). Only after learning Algebra and homomorphisms, isomorphisms and automorphisms did I appreciate the importance of linear transformations. Stuff like Singular Value…

That was exactly my experience. Struggled with matrices theory at uni doing some bullshit exercises but started to grasp the topic only when I needed to apply some linear transformation in a game

Re: Relearning Matrices as Linear Functions

#49

Post by wolfgke » Tue, Apr 23, 2019, 10:11 PM UTC

throwawaymath wrote: post #43

wolfgke wrote: post #26

Earlier quoted context omitted.

Shearings cannot be represented in this way.

Yes they can. This follows from singular value decomposition. Let S be the matrix representation of a shear transformation. There exist rotation matrices R, B and a diagonal matrix D such that S = RDC , where C is the transpose of B . D is the matrix representation of a scaling transformation and R, B are the matrix representations of rotation transformations. Since S is a product of rotation and scaling matrices, it…

OK, point taken. I considered "scaling" in a less general sense (scalar multiple of the unit matrix), while you want to allow arbitrary diagonal entries. My definition is to my knowledge the common one in linear algebra textbooks because in yours, the feasible maps depend on the chosen basis.

EDIT: To state my point more clearly: in textbooks, "scaling" is the linear map that is induced by the "scalar multiplication" in the definition of the vector space (that is why both terms start with "scal").

Re: Relearning Matrices as Linear Functions

#50

Post by btilly » Tue, Apr 23, 2019, 10:13 PM UTC

whatshisface wrote: post #2

FWIW, I was told that matrices are linear maps pretty early on in my education. Are there any college level linear algebra / matrix calculations courses that don't tell students about that?

Sadly, there are. Or at least were.

When I went through university the standard set of courses was a Calculus course that was mostly about derivatives, a second one that was mostly about integrals, a third Calculus course that was about multi-variable Calculus. That third course necessarily had to teach matrices, and taught it as rote calculations. There was a follow-up differential equations course which refreshed people's memories of matrices..as a rote calculation.

It was done this way because the multi-variable Calculus course was a prerequisite for a lot of physics+engineering courses. So a lot of students wanted to take that sequence. Differential equations were a prerequisite for some other advanced courses. Linear algebra was pretty much just for math majors.