
Affine transformations
Hi, the complete original question is:
f and g are both affine transformations. The transformation f is a reflection in the line x=2, and the transformation g maps the points (0,0), (1,0) and (0,1) to the points (0, 4), (0, 3) and (1, 4) respectively.
a) determine g (in the form g(x) Ax + a, where A is a 2x2 matrix and a is a vector with two components.
b) Exprerss f as a composite of three transformations: a translation, followed by a 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)).
c) Fine the affine transformation g o f (in the same form as you found g in part a)).
d) Given that g o f is either a rotation or a reflection, state which it is. If it is a rotation, state the centre and angle of rotation. If it is a reflection, state the axis of reflection. Justify your answer.
For a) I found that (g = Bx + b) g(x) =
$\displaystyle \left(\begin{array}{cc}0&1\\1&0\end{array}\right)$ + $\displaystyle \left(\begin{array}{cc}0\\4\end{array}\right)$
For b) I found that (f = Ax + a) f(x) =
$\displaystyle \left(\begin{array}{cc}1&0\\0&1\end{array}\right)$ + $\displaystyle \left(\begin{array}{cc}4\\0\end{array}\right)$
c) I am unsure how to combine the two answers above. I thought something like: (?)
g o f = g(f(x))
= g(Ax + a)
= B(Ax + a) + b
d) Also I am not sure how to work out this question.
Any help would be much appreciated. Thanks for your help in advance.
Kind regards,
Cozza

stuck too
hi cozza
did you get this
im doing it as part of a OU tma
stuck on part c aswell

This is what my tutor said:
g(f(x)) = g(Ax + a)
= B(Ax + a) + b
You are absolutely right so far. Now you need to multiply out the bracket
just as if you were working with algebra except here you will be
multiplying matrices & vectors together. You will then end up with
something in the form Cx + c.
I can't point you to a similar example in the text as I know there isn't
one!
I have multiplied out the bracket (I hope it's right!) ansd got an answer in the form Cx + c, but I'm not sure about part d) now.
Hope that helps. Let me know if you have any further luck

i got for a: 0 1 x + 0
_________ 1 0_____4
then for b i have used reflection in y axis matrix: 1 0
__________________________________________0 1
and got ans b to be: 1 0 x + 0
___________________0 1____4
then ans c i got: 1 0 x + 4
_______________01____4
?
what are you using for reflection matrix?

Hi for part (c) I understand and end up with this below,
g o f = Cx + c
C = 0 1 and c = 0
1 0 0
You can calculate this by multiplying the matrix for g with the matrix for f, so
C = A*B and c = A*b + a
Its just the part (b) that is really throwing me now as how I got to here (part c)