hi cozza
did you get this
im doing it as part of a OU tma
stuck on part c aswell
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) =
+
For b) I found that (f = Ax + a) f(x) =
+
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
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
_______________0-1____-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)