I can understand maybe 40 percent. What is your limit? What subject pushes you to the limit?
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Yeah, I donât even know what this is.This stuff goes way over my head
Yeah, there is a way to break things down more, but nobody's perfect I guess.I donât think Iâd have explained any of that the way he did, bouncing between the concepts a little too freely for my liking. But following it wasnât 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, youâre really constructing a box labelled f that has an in-tray and an out-tray, if you put 2 in, youâll get 4 out. Put 20 in, youâll 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 weâre 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. Heâs got a fancy grid where itâs 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.
Heâs 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 âI want to find 1,0 on the planeâ, itâs now at 2,1 - thereâs your linear transformation at work.
What he then goes on to do is go âI have these arrows, but Iâm going to use vector notation, deal with itâ to then go âok, so Iâve done a transformation and by smashing together these two vectors into a matrix I describe the transformationâ 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 âjourneys from a reference pointâ. Just as you can describe âgoing 1 mile east from your front door, no miles northâ, 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 â4 numbersâ he keeps talking about, which is really just one way of âwriting a function that says when you ask me for this, I give you thatâ to deal with âhey I rotated the world, where things at nowâ
The âlinearâ 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 donât do anything that multiplies the inputs by themselves even by accident. Thereâs no âx squaredâ or âx to the power ofâ in a linear function.
I said earlier that you all drew y = x^2 in school. Iâm sure you also drew y = 2x as a straight line going upwards twice as fast as it went along. Well, what youâre 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 itâs a *straight line*.
Thatâs why itâs 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.
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