# Affine transformations

• May 31st 2010, 10:06 AM
cozza
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) =
$\left(\begin{array}{cc}0&-1\\1&0\end{array}\right)$ + $\left(\begin{array}{cc}0\\-4\end{array}\right)$

For b) I found that (f = Ax + a) f(x) =
$\left(\begin{array}{cc}-1&0\\0&1\end{array}\right)$ + $\left(\begin{array}{cc}4\\0\end{array}\right)$

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
• Jun 3rd 2010, 02:30 AM
samson1
stuck too
hi cozza
did you get this
im doing it as part of a OU tma
stuck on part c aswell
• Jun 3rd 2010, 02:55 AM
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
• Jun 3rd 2010, 03:43 AM
samson1
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?
• Jun 3rd 2010, 07:15 AM
Tinca18
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)