The Limit of Understanding

[Jason]

I can understand maybe 40 percent. What is your limit? What subject pushes you to the limit?

 

[Yeti]

This stuff goes way over my head

 

[Cedric]

This stuff goes way over my head

Yeah, I don’t even know what this is.

 

[Debashis]

There are easier ways to explain linear algebra. His videos are not meant for an audience with <= average knowledge of math.
If you watch the below video, I'm sure you'll understand almost all of it. And maybe you'll be able to understand the video in the thread more too.

 

[Jason]

I think his videos are cool and people should know a deep understanding of the subject, but I doubt if most people do. Even myself, I can look at patterns and pass tests, but it's kind of a con-job really.

 

[Ravenfreak]

Math goes over my head, I always struggled with it and still struggle with things as simple as fractions. :S Yet it's funny, when it comes to coding and modifying games I don't have a problem with that and it all comes down to logic for both. :V I can easily start modifying existing code if I study it a little bit.

 

[Jason]

A big problem with this stuff is I just don't get it. It's just words here and there. I'm sure you all are in the same boat also. I wonder how to get it.

 

[Arantor]

I don’t think I’d have explained any of that the way he did, bouncing between the concepts a little too freely for my liking. But following it wasn’t a challenge at all.

So, the quick recap: a function is a box that has some inputs and some outputs. When you say f(x) = x^2, you’re really constructing a box labelled f that has an in-tray and an out-tray, if you put 2 in, you’ll get 4 out. Put 20 in, you’ll get 400 out.

Next aspect: a transformation is a function that materially alters the input. The f(x) box above takes in x and gives you back something not x.

Stepping aside from the pure algebra side for a bit, any function that accepts a number and spits out a number can be drawn on a graph. I am certain you all drew the graph of y = x^2 in school - it looks like a sort of U shape, where when x is 1, y is 1; x is 2, y is 4 and so on.

All we’re doing then is wrapping up that into a generic box.

The next step in his logic is to deal with the concept of a plane. A plane is a flat thing with a centre and you have coordinates that go in all directions from that centre. He’s got a fancy grid where it’s marked all of the divisions from the centre in units of 1, so theres a marker point at 1,1; one at 2,1; one at 3,1 and so on, then just lines drawn between them.

Then what he does, and fails to explain, is that he has created a transform that applies to every single point on the plane at once.

He’s got his little function f(x2, y2) = (something that happens to x1, y1) and applied it to the plane. The result is that if you say “I want to find 1,0 on the plane”, it’s now at 2,1 - there’s your linear transformation at work.

What he then goes on to do is go “I have these arrows, but I’m going to use vector notation, deal with it” to then go “ok, so I’ve done a transformation and by smashing together these two vectors into a matrix I describe the transformation” without stopping to explain *why*.

So, he describes the yellow arrow and the red arrow. These are two separate things. This is extremely badly explained.

The arrows are “journeys from a reference point”. Just as you can describe “going 1 mile east from your front door, no miles north”, you can describe going 1,0 from the origin or centre of the plane.

You put the red arrow into the box, so in goes (1,0), out comes (2,1).
You put the yellow arrow into the box, so in goes (0,-1), out comes (-1,1)

The nuts and bolts are that you can describe what the box is doing in every single case for every single possible point on the map as a matrix manipulation, which is the “4 numbers” he keeps talking about, which is really just one way of “writing a function that says when you ask me for this, I give you that” to deal with “hey I rotated the world, where things at now”

The “linear” part was also not well explained. Transformations can occur in all sorts of ways - linear is the easiest: you have a box that takes in something, you do some multiplications, divisions, additions, but you don’t do anything that multiplies the inputs by themselves even by accident. There’s no “x squared“ or “x to the power of” in a linear function.

I said earlier that you all drew y = x^2 in school. I’m sure you also drew y = 2x as a straight line going upwards twice as fast as it went along. Well, what you’re doing there is applying a linear function - x is 1, y must be 2, you draw the dot at 1,2; x is 2, y is 4, you draw the dot at 2,4; x is 3, y is 6, you draw the dot at 3,6… then you join the dots and it’s a *straight line*.

That’s why it’s a linear function: because if you draw out its result as a graph you get a straight line.

x^2 is not linear - if you draw it out (applying the function as you draw each dot) it gets steeper and steeper the higher you go.
sin x is also not linear, it makes a lovely up and down curve as does cosine.

Beyond that, really once you get your head around the idea of a function, a transformation is just a function applied everywhere all at once, a matrix manipulation is just one way to describe such a function.

And programming is all just algebra and calculus in the end.

 

[Jason]

I don’t think I’d have explained any of that the way he did, bouncing between the concepts a little too freely for my liking. But following it wasn’t a challenge at all.

So, the quick recap: a function is a box that has some inputs and some outputs. When you say f(x) = x^2, you’re really constructing a box labelled f that has an in-tray and an out-tray, if you put 2 in, you’ll get 4 out. Put 20 in, you’ll get 400 out.

Next aspect: a transformation is a function that materially alters the input. The f(x) box above takes in x and gives you back something not x.

Stepping aside from the pure algebra side for a bit, any function that accepts a number and spits out a number can be drawn on a graph. I am certain you all drew the graph of y = x^2 in school - it looks like a sort of U shape, where when x is 1, y is 1; x is 2, y is 4 and so on.

All we’re doing then is wrapping up that into a generic box.

The next step in his logic is to deal with the concept of a plane. A plane is a flat thing with a centre and you have coordinates that go in all directions from that centre. He’s got a fancy grid where it’s marked all of the divisions from the centre in units of 1, so theres a marker point at 1,1; one at 2,1; one at 3,1 and so on, then just lines drawn between them.

Then what he does, and fails to explain, is that he has created a transform that applies to every single point on the plane at once.

He’s got his little function f(x2, y2) = (something that happens to x1, y1) and applied it to the plane. The result is that if you say “I want to find 1,0 on the plane”, it’s now at 2,1 - there’s your linear transformation at work.

What he then goes on to do is go “I have these arrows, but I’m going to use vector notation, deal with it” to then go “ok, so I’ve done a transformation and by smashing together these two vectors into a matrix I describe the transformation” without stopping to explain *why*.

So, he describes the yellow arrow and the red arrow. These are two separate things. This is extremely badly explained.

The arrows are “journeys from a reference point”. Just as you can describe “going 1 mile east from your front door, no miles north”, you can describe going 1,0 from the origin or centre of the plane.

You put the red arrow into the box, so in goes (1,0), out comes (2,1).
You put the yellow arrow into the box, so in goes (0,-1), out comes (-1,1)

The nuts and bolts are that you can describe what the box is doing in every single case for every single possible point on the map as a matrix manipulation, which is the “4 numbers” he keeps talking about, which is really just one way of “writing a function that says when you ask me for this, I give you that” to deal with “hey I rotated the world, where things at now”

The “linear” part was also not well explained. Transformations can occur in all sorts of ways - linear is the easiest: you have a box that takes in something, you do some multiplications, divisions, additions, but you don’t do anything that multiplies the inputs by themselves even by accident. There’s no “x squared“ or “x to the power of” in a linear function.

I said earlier that you all drew y = x^2 in school. I’m sure you also drew y = 2x as a straight line going upwards twice as fast as it went along. Well, what you’re doing there is applying a linear function - x is 1, y must be 2, you draw the dot at 1,2; x is 2, y is 4, you draw the dot at 2,4; x is 3, y is 6, you draw the dot at 3,6… then you join the dots and it’s a *straight line*.

That’s why it’s a linear function: because if you draw out its result as a graph you get a straight line.

x^2 is not linear - if you draw it out (applying the function as you draw each dot) it gets steeper and steeper the higher you go.
sin x is also not linear, it makes a lovely up and down curve as does cosine.

Beyond that, really once you get your head around the idea of a function, a transformation is just a function applied everywhere all at once, a matrix manipulation is just one way to describe such a function.

And programming is all just algebra and calculus in the end.

Yeah, there is a way to break things down more, but nobody's perfect I guess.