Relearning Matrices as Linear Functions • Hacker News Board

Re: Relearning Matrices as Linear Functions

#91

Post by mikorym » Wed, Apr 24, 2019, 12:51 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)?

So yes, this is equivalent to FinVect of the field of which entries in the matrices consist.

The difference is that here you construct the category from a simpler premise. To construct FinVect you need to include all set objects with structure satisfying some axioms.

The category of matrices is simply positive integers with as morphisms n x m matrices between the two integers. Composition is matrix multiplication.

Here [1] is a nice overview. If you can follow what is going on there, it is worth while looking at II, III and IV.

[1] https://unapologetic.wordpress.com/2008/06/02/the-category-o...

Re: Relearning Matrices as Linear Functions

#92

Post by bonoboTP » Wed, Apr 24, 2019, 3:04 PM UTC

munchbunny wrote: post #37

bonoboTP wrote: post #35

Earlier quoted context omitted.

You're talking about translation.

No, I wasn't, but I did confuse the terms. Shear can be done without the extra dimension. Skew transforms require the extra dimension, as does translation.

What do you mean by skew? A perspective transformation (homography)? I'm not sure it's standard terminology.

Re: Relearning Matrices as Linear Functions

#93

Post by p1esk » Wed, Apr 24, 2019, 5:15 PM UTC

soVeryTired wrote: post #83

p1esk wrote: post #65

Earlier quoted context omitted.

That didn’t make any sense to, and I work with matrices every day. Are you trying to describe a dot product?

I'm describing a somewhat unusual way of thinking about vectors, matrices etc. At least, it's unusual from the perspective of someone with an engineering / CS background. First think about row and column vectors. A row vector and a column vector can be combined via standard matrix multiplication to produce a real number. From that perspective, a row vector is a function that takes a column vector and returns a real n…

I appreciate you trying to explain it, however I believe it would have really helped if you started from the examples where this way of thinking is useful: you mentioned tensor product and vector space duality, however unless someone is already familiar with these concepts (I'm not) then it does seem needlessly convoluted. Are there any practical applications of these concepts that you can describe?

Re: Relearning Matrices as Linear Functions

#94

Post by je42 » Wed, Apr 24, 2019, 6:16 PM UTC

this was an important result in the linear algebra class for first year math/cs/eng students at my university.

Re: Relearning Matrices as Linear Functions

#95

Post by kaitai » Sat, Apr 27, 2019, 6:59 PM UTC

soVeryTired wrote: post #56

dhruvp wrote: post #15

Hey OP here! When I first was introduced to matrices (high school) it was in the context of systems of equations. Matrices were a shorthand for writing out the equations and happened to have interesting rules for addition etc. It took me a while to think about them as functions on their own right and not just tables. This post is my attempt to relearn them as functions which has helped me develop a much stronger intu…

It's a nice article - you focus on matrices as a kind of operator that takes a vector as input and produces another vector. This is one side of the coin. The other interpretation is that matrices are functions that take two arguments (a row vector and a column vector) and produce a real number. IMO this interpretation opens the door to deeper mathematics. It links in to the idea that a column vector is a functional o…

If you're going to attempt a definition like this you need some more conditions (approaching concepts like linearity, for instance). Otherwise you can have a black box like x^3y^3-u^3v^3 that takes in [u,v] and [x,y]^T and spits out a real number; that's not a matrix-y operation.