Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

The Little Book of Linear Algebra

github.com

Post reply on HN
Print view View original

111–120 of 134 posts

Re: The Little Book of Linear Algebra

#111

Post by ferfumarma »

andrewla wrote: post #40
srean wrote: post #17

Earlier quoted context omitted.

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away In my experience it need not be like that at all. One can start by defining and demonstrating linear transformations. Perhaps from graphics -- translation, rotation, reflection etc. Show the students that these fol…

> Perhaps from graphics -- translation, rotation, reflection Maybe ... but the fact that you included translation in the list of linear operations seems like a big red flag. Translation feels very linear but it is emphatically not [1]. This is not intended to be a personal jab; just that the intuitions of linear algebra are not easy to internalize. Adding linear transformations is similarly scary territory. You can m…

Can you elaborate on your point that translation is not linear? The OP agrees with you, so clearly your point is correct, but I personally just don't understand it. Isn't it true that translation is linear within the coordinate space of your model, even if the final distance traveled within a projected camera view is not?

edit to add: (I think your point relates only to the projection system, and not a pure, unprojected model; I just want to make sure I understand because it seems like an important point)

Re: The Little Book of Linear Algebra

#112

Post by srean »

ferfumarma wrote: post #111
andrewla wrote: post #40

Earlier quoted context omitted.

> Perhaps from graphics -- translation, rotation, reflection Maybe ... but the fact that you included translation in the list of linear operations seems like a big red flag. Translation feels very linear but it is emphatically not [1]. This is not intended to be a personal jab; just that the intuitions of linear algebra are not easy to internalize. Adding linear transformations is similarly scary territory. You can m…

Can you elaborate on your point that translation is not linear? The OP agrees with you, so clearly your point is correct, but I personally just don't understand it. Isn't it true that translation is linear within the coordinate space of your model, even if the final distance traveled within a projected camera view is not? edit to add: (I think your point relates only to the projection system, and not a pure, unprojec…

All linear operators map origin to origin. But translation applied to the origin will shift it. So translation cannot be linear.

Let's take another approach.

Take a point p that's sum of vectors a and b, that is

p = a + b.

Now, if translation was a linear transformation, then translating p (say along x-axis by 1 unit) is equivalent to applying same translation to a and b separately and then summing them. But the latter ends up translating by twice the amount. Or in other words

p +t ≠ (a +t) + (b +t) = p + 2t.

So translation is not a linear operators in this vector space.

Re: The Little Book of Linear Algebra

#113

Post by srean »

imtringued wrote: post #108
srean wrote: post #17

Earlier quoted context omitted.

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away In my experience it need not be like that at all. One can start by defining and demonstrating linear transformations. Perhaps from graphics -- translation, rotation, reflection etc. Show the students that these fol…

>The beautiful reveal is that this addition and composition of linear transformations behave almost the same as addition and multiplication of real numbers. This is only beautiful if you already understand monoids, magmas and abelian half groups (semigroups) and how they form groups. Also, we do not talk of linear transformations, we talk of group homomorphisms. I don't know about anyone else, but I was taught linear…

It can be beautiful with less.

All that needs to be demonstrated is that for real numbers + associates and commutes. That * associates and commutes. And most satisfyingly, these two operations interact through the distribution property.

Of course, it's more revealing and interesting if one has some exposure to groups and fields.

Do people encounter linear algebra in their course work before that ?

For us it came after coordinate/analytical geometry where we had encountered parallelogram law. So while doing LA we had some vague awareness that there's a connection. This connection solidified later.

We also had an alternative curriculum where matrices were taught in 9th grade as a set of rules without any motivation whatsoever. "This is the rule for adding, this one's for multiplication, see you at the test"

Re: The Little Book of Linear Algebra

#116

Post by akww »

eliaskickbush wrote: post #58

Highly recommend to anyone struggling with linear algebra to check out Linear Algebra Done Right, by Sheldon Axler. Do always keep in mind that some concepts are very verbose, but truly out of necessity. If you're talking about an N by N matrix, you're naturally going to have to distinguish N^2 different elements. You can go very far without touching matrices, and actually find motivation on this abstract base before…

A little surprising to me this doesn’t come up the most! Excellent text, and look, you can get the 4th edition (2024) for free as a pdf at http://axler.net.

Re: The Little Book of Linear Algebra

#117

Post by nh23423fefe »

jcranmer wrote: post #70
nh23423fefe wrote: post #61

Earlier quoted context omitted.

I dont agree with this. Matrices don't convert sets of basis vectors to sets of basis vectors. What would you say about singular matrices for example? The natural motivation of matrices is as representing systems of equations.

If I write a vector v = [1, 3, 2], what I am actually saying is that v is equal to 1 * e₁ + 3 * e₂ + 2 * e₃ for three vectors I have previously decided on ahead of time that form an orthonormal basis of the corresponding vector space. If I write a matrix, say, this: [[1 2] [3 4] [5 6]] What I am doing is describing is a transformation of one vector space into another, by describing how the basis vectors of the first…

Doesn't make sense is too strong though.

If you have a system Ax=y and a system By=z there exists a system (BA)x=z

This system BA is naturally seen as the composition of both systems of equations

And the multiplication rule expresses the way to construct the new systems' coefficients over x constrained by z.

The C_i equation has coefficients which are the evaluations of the B_i equation over the A_k-th coefficients

C_ik = B_ij A_jk

concretely

        A11 x1 + A12 x2 = y1
        A21 x1 + A22 x2 = y2

        and

        B11 y1 + B12 y2 = z1
        B21 y1 + B22 y2 = z2

        then

        B11 (A11 x1 + A12 x2) + B12 (A21 x1 + A22 x2) = z1
        B21 (A11 x1 + A12 x2) + B22 (A21 x1 + A22 x2) = z2

        rearrange and collect terms

        (B11 A11 + B12 A21) x1 + (B11 A12 + B12 A22) x2 = z1
        (B21 A11 + B22 A21) x1 + (B21 A12 + B22 A22) x2 = z2
the coefficients express the dot product rule directly

Re: The Little Book of Linear Algebra

#119

Post by tamnd »

mugamuga wrote: post #95

Is there a version or a similar book that deals with Calculus?

[Author here] We hear you! Here's a similar book for Calculus: https://github.com/the-little-book-of/calculus

In this book, I cover Functions, Derivatives, Integrals, Multivariable Calculus, and Infinite Processes. In addition, I've included appendices with sketch proofs and applications to Physics, Probability and Statistics, and Computer Science.

Re: The Little Book of Linear Algebra

#120

Post by tamnd »

[Author here] Linear Algebra is the first book in a series I'm writing on math, computer science, and programming. You can find the index for the upcoming books here: https://github.com/the-litte-book-of/everything. If the style clicks with you or you're curious about certain topics, I’d love to hear your thoughts!
Post reply on HN