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Matrices can be your friends (2002)

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Re: Matrices can be your friends (2002)

#31

Post by treyd »

foofoo12 wrote: post #19
omnicognate wrote: post #13

Earlier quoted context omitted.

A lot of people who find themselves having to deal with matrices when programming have never taken that class or learned those things (or did so such a long time ago that they've completely forgotten). I assume this is aimed at such people, and he's just reassuring them that he's not going to talk about the abstract aspects of linear algebra, which certainly exist. I'd take issue with his "most programmers are visual…

> most programmers are visual thinkers I remember reading that there's a link between aphantasia (inability to visualize) and being on the spectrum. Being an armchair psychologist expert with decades of experience, I can say with absolute certainty that a lot of programmers are NOT visual thinkers.

This is interesting because, to me, programing is a deeply visual activity. It feels like wandering around in a world of forms until I find the structures I need and actually writing out the code is mostly a formality.

Re: Matrices can be your friends (2002)

#32

Post by card_zero »

ndriscoll wrote: post #24
card_zero wrote: post #18

Earlier quoted context omitted.

OK, now what?

A matrix is just a list of where a linear map sends each basis element (the nth column of a matrix is the output vector for the nth input basis vector). Lots of things are linear (e.g. scaling, rotating, differentiating, integrating, projecting, and any weighted sums of these things). Lots of other things are approximately linear locally (the derivative if it exists is the best linear approximation. i.e. the best mat…

Yes, I think of them as saying "and this is what the coordinates in our coordinate system [basis] shall mean from now on". Systems of nonlinear equations, on the other hand, are some kind of sea monsters.

Re: Matrices can be your friends (2002)

#33

Post by whizzter »

getnormality wrote: post #6

People must get taught math terribly if they think "I don't need to worry about piles of abstract math to understand a rotation, all I have to do is think about what happens to the XYZ axes under the matrix rotation". That is what you should learn in the math class! Anyone who has taken linear algebra should know that (1) a rotation is a linear operation, (2) the result of a linear operation is calculated with matrix…

Honestly, many math teachers are kinda bad at conveying all that.

When everything clicked a few years down the line it all became so simple.

Like you mention "linear operation", the word linear doesn't always make intuitive sense in terms of rotations or scaling if you have encountered simple 1 or 2 dimensional linear transformations when doing more basic graphics programming.

As a teacher, I think the biggest lesson I had to learn was to always have at least 3 different ways of explaining everything to give different kinds of people different entrypoints into understanding concepts.

For someone uninitiated a term like "basis vector" can be pure gibberish if it doesn't follow an example of a transform as a viewport change, and it needs to be repeated after your other explanations (of for example how vector components in the source view just are scalars upon the basis vectors when multiplied with a matrix instead of a heavy un-intuitive concept).

Re: Matrices can be your friends (2002)

#34

Post by zkmon »

blackbear_ wrote: post #26
zkmon wrote: post #12

There are a lot more ways to look at and understand these mysterious beasts called matrices. They seem to represent a more fundamental primordial truth. I'm not sure what it is. Determinant of a matrix indicate the area of or volume spanned by its component vectors. Complex matrices used in Fourier transform are beautiful. Quantum mechanics and AI seem to be built on matrices. There is hardly any area of mathematics…

The fundamental truth is that matrices represent linear transformations, and all of linear algebra is developed in terms of linear transformations rather than just grid of numbers. It all becomes much clearer when you let go of the tabular representation and study the original intentions that motivated the operations you do on matrices. My appreciation for the subject grew considerably after working through the book…

Spatial transformations? Take a look at the complex matrices in Fourier transforms with nth roots of unity as its elements. The values are cyclic, and do not represent points in an n-D space of Euclidean coordinates.

Re: Matrices can be your friends (2002)

#35

Post by notmyjob »

foofoo12 wrote: post #19
omnicognate wrote: post #13

Earlier quoted context omitted.

A lot of people who find themselves having to deal with matrices when programming have never taken that class or learned those things (or did so such a long time ago that they've completely forgotten). I assume this is aimed at such people, and he's just reassuring them that he's not going to talk about the abstract aspects of linear algebra, which certainly exist. I'd take issue with his "most programmers are visual…

> most programmers are visual thinkers I remember reading that there's a link between aphantasia (inability to visualize) and being on the spectrum. Being an armchair psychologist expert with decades of experience, I can say with absolute certainty that a lot of programmers are NOT visual thinkers.

Math achievement correlates strongly with visuospatial reasoning. Programmers may not be as proficient in math as economists, but they are better at it than biologists or lawyers.

Re: Matrices can be your friends (2002)

#36

Post by FabHK »

knappa wrote: post #30
Viliam1234 wrote: post #5

> Mathematicians like to see their matrices laid out on paper this way (with the array indices increasing down the columns instead of across the rows as a programmer would usually write them). Could a mathematician please confirm of disconfirm this? I think that different branches of mathematics have different rules about this, which is why careful writers make it explicit.

I'm a mathematician. It's kind of a strange statement since, if we are talking about a matrix, it has two indices not one. Even if we do flatten the matrix to a vector, rows then columns are an almost universal ordering of those two indices and the natural lexicographic ordering would stride down the rows.

Yes. I think what all mathematicians can agree on is that the layout (and the starting index! :-) is like this:

  A[1,1] … A[1,n]
   …        …
  A[m,1] … A[m,n]

Re: Matrices can be your friends (2002)

#38

Post by nyrikki »

dgacmu wrote: post #17
zelphirkalt wrote: post #15

Earlier quoted context omitted.

When I was studying and made the mistake of choosing 3D computer graphics as a lecture, I remember some 4x4 matrix that was used for rotation, with all kinds of weird terms in it, derived only once, in a way I was not able to understand and that didn't relate to any visual idea or imagination, which makes it extra hard for me to understand it, because I rely a lot on visualization of everything. So basically, there w…

You can do rotation with a 3x3 matrix. The first lecture was using a 4x4 matrix because you can use it for a more general set of transformations, including affine transforms (think: translating an object by moving it in a particular direction). Since you can combine a series of matrix multiplications by just pre-multiplying the matrix, this sets you up for doing a very efficient "move, scale, rotate" of an object usi…

> You can do rotation with a 3x3 matrix.

You can do a rotation or some rotations but SO(3) is not simply connected.

It mostly works for rigid bodies centered on the origin, but gimbal lock or Dirac's Plate Trick are good counter example lenses. Twirling a baton or a lasso will show that 720 degrees is the invariant rotation in SO(3)

The point at infinity with a 4x4 matrix is one solution, SU(3), quaternions, or recently geometric product are other options with benefits at the cost of complexity.

Re: Matrices can be your friends (2002)

#39

Post by blackbear_ »

zkmon wrote: post #34
blackbear_ wrote: post #26

Earlier quoted context omitted.

The fundamental truth is that matrices represent linear transformations, and all of linear algebra is developed in terms of linear transformations rather than just grid of numbers. It all becomes much clearer when you let go of the tabular representation and study the original intentions that motivated the operations you do on matrices. My appreciation for the subject grew considerably after working through the book…

Spatial transformations? Take a look at the complex matrices in Fourier transforms with nth roots of unity as its elements. The values are cyclic, and do not represent points in an n-D space of Euclidean coordinates.

Yes; I wrote linear transformation on purpose not to remain constrained on spatial or geometric interpretations.

The (discrete) Fourier transform is also a linear transformation, which is why the initial effort of thinking abstractly in terms of vector spaces and transformations between them pays lots of dividends when it's time to understand more advanced topics such as the DFT, which is "just" a change of basis.

Re: Matrices can be your friends (2002)

#40

Post by whizzter »

bmacho wrote: post #20
zkmon wrote: post #12

There are a lot more ways to look at and understand these mysterious beasts called matrices. They seem to represent a more fundamental primordial truth. I'm not sure what it is. Determinant of a matrix indicate the area of or volume spanned by its component vectors. Complex matrices used in Fourier transform are beautiful. Quantum mechanics and AI seem to be built on matrices. There is hardly any area of mathematics…

A matrix is just a grid of numbers. A lot of areas use use grid of numbers. And matrix theory actually incorporates every area that uses grids of numbers, and every rule in those areas. For example the simplest difficult thing in matrix theory, matrix multiplication is an example for this IMO. It looks really weird in the context of grid of numbers, and its properties seem incidental, and the proofs are complicated.…

This is the most important part.

"...linear transformations between vector spaces."

When you understand what that implies you can start reasoning about it visually.

The 3 simplest (that you can find in blender or any other 3d program, or even partly in 2d programs).

Translation (moving something left,right,up,down,in,out).

Rotation (turn something 2 degrees, 90 degrees, 180 degrees, 360 degrees back to the same heading)

Scaling (make something larger, smaller, etc)

(And a few more that doesn't help right now)

The 2 first can be visualized simply in 2d, just take a paper/book/etc. Move it left-right, up down, rotate it.. the book in the original position and rotation compared to the new position and rotation can be described as a vector space transformation, why?

Because you can look at it in 2 ways, either the book moved from your vantage point, or you follow the book looking at it the same way and the world around the book moved.

In both cases, something moved from one space (point of reference) to another "space".

The thing that defines the space is a "basis vector", basically it says what is "up", what is "left" and was in "in" in the way we move from one space to another.

Think of it as, you have a piece card on a paper. Draw an line/axis along the bottom edge as the X count, then draw on the left side upwards the Y count. In the X,Y space (from space) you count the X and Y steps of various feature points.

Now draw the "to space" as another X axis and another Y axis (could be rotated, could be scaled, could just be moved) and take the counts in steps and put them inside the "to space" measured in equal units as they were in the from space.

Once the feature points are replicated in the "to space" you should have the same image as before, just within the new space.

This is the essence of a so called linear(equal number steps) transform (moved somewhere else), and also exactly what multiplying a set of vectors by a matrix achieves (simplified, in this context, the matrix really is mostly a representation of a number of above mentioned basis vectors that defines the X, Y,etc of the movement).

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