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February 1st, 2012, 01:39 AM   #1
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Linear and affine transformations

Hey all! hope all is well! this question is quite long, and leads on from one another, i'm just stuck on the second part, if anyone could explain thanks!

In this question, f and g are both affine transformations. The
transformation f is reflection in the line y = 2, and the transformation g
maps the points (0, 0), (1, 0) and (0, 1) to the points (1, 1), (2, 2) and
(3,?1), respectively.

(a) Determine g (in the form g(x) = Ax + a, where A is a 22 matrix
and a is a vector with two components).

Done this part which equals g(x) = (matrix) x + (vector) the matrix is (a,b,c,d) (1,2,1,-2) vector (a,b) (1,1)

the next part i don't understand

(b) Express f as a composite of three transformations: a translation,
followed by reflection in a line through the origin, followed by a
translation. Hence determine f (in the same form as you found g in
part (a)).

I understand what u have to do here and the process u need to take, but i can't understand if y=2 then surely it will be parallel with the x axis, which wont make an angle which u need for the formula of a reflection matrix (a = cos(2phi) b= sin(2phi) c= sin(2phi) d= -cos(2phi))
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February 11th, 2012, 06:30 AM   #2
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Re: Linear and affine transformations

hey,had the same problem,from what ive worked out you need to translate back onto the x axix,then reflect through the x-axis,and then undo your first translation by moving the line again,so i got:

(x,y) ? (x,y-2)
(x,y) ? (x,-y)
(x,y) ? (x,y+2)

but i am not sure how to express this as 1 composite function,i know each element individually but how to combine it i dont know so if you do them please lemme know would be appreciated,cheers)
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February 17th, 2012, 03:48 PM   #3
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Re: Linear and affine transformations

hey nmenumber1 so sorry about the late reply!

First off thanks for your help!,

I think the question is very misleading! took me some time to figure it out!

But I finally found out what to do! hope this helps! if you've already solved it then great but this is what i got!

To compose this

as a composite of three transformations, (a translation, followed by a reflection, followed by a translation) using matrices i got

[[(X) + (0, -2)] multiply by [1,0,0,-1]] + (0,2)

so the first part would be the translation, involving a point x to be translated to the x-axis [(X) + (0, -2)]

the second part would be the reflection [1,0,0,-1] - matrices [ a,b,c,d ]

and the third part would be the translation back to the originally point + (0,2)

as we want this in the form f(x) = A x + a

We write

f: R^2 (arrow pointing right) R^2

f(x) = (1,0,0,-1)(x+(0,-2)) + (0,2)

Hope this helps any other help please ask!, and i will get to you as soon as possible ! a lot faster!

This can also be written like

f(x) = A(x-a) +a

f: R^2 (arrow pointing right) R^2

f(x) = (1,0,0,-1)(x+(0,2)) + (0,2)
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February 17th, 2012, 03:51 PM   #4
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Re: Linear and affine transformations

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