Affine transformation question...HELP!

Feb 2014
54
2
England
In this question,
f and g are both affine transformations. The
transformation
f is reflection in the line y = x 1, and the transformation
g
maps the points (0, 0), (1, 0) and (0, 1) to the points (3,1), (4,1) and
(3,-2), respectively.
(a) Determine
g in the form g(x) = Ax + a, where A is a 2×2 matrix
and
a is a vector with two components.

(b) Express
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).

I have answers to these questions but I'm really not sure if they're right, so if someone could put up the correct answers so I can see how I did then that would be great! Thanks :)



 

romsek

MHF Helper
Nov 2013
6,666
3,003
California
In this question,
f and g are both affine transformations. The
transformation
f is reflection in the line y = x − 1, and the transformation
g
maps the points (0, 0), (1, 0) and (0, 1) to the points (3,−1), (4,−1) and
(3,-2), respectively.
(a) Determine
g in the form g(x) = Ax + a, where A is a 2×2 matrix
and
a is a vector with two components.

(b) Express
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).
a)

A(0,0) + a=(3,-1) so clearly

$a=\left(\begin{array}{c}3\\-1\end{array}\right)$

A(1,0) + (3,-1) = (4,-1)
A(1,0) = (1,0)

A(0,1) + (3,-1) = (3,-2)
A(0,1) = (0,-1)

$A =\left(\begin{array}{cc}1&0\\0 &-1\end{array}\right)$

b)

$t=\left(\begin{array}{c}0\\-1\end{array}\right)$

$r=\left(\begin{array}{cc}0 &1\\1 &0\end{array}\right)$

$f(x)=r(x-t)+t$
 
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Feb 2014
54
2
England
First, thanks for replying!
My answer to part a is definitely right, and for part b I get (in the form (a,b,c,d) for the matrix)

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

Please could you tell me if that's right too, otherwise my answers to the next part of the question will be wrong?!
 
Last edited:

romsek

MHF Helper
Nov 2013
6,666
3,003
California
First, thanks for replying!
My answer to part a is definitely right, and for part b I get (in the form (a,b,c,d) for the matrix)

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

Please could you tell me if that's right too, otherwise my answers to the next part of the question will be wrong?!
I posted the correct form for f(x). What you have is incorrect. You neglect the first translation.
 

romsek

MHF Helper
Nov 2013
6,666
3,003
California
Ok I'm wrong. When you take my expression and expand it out it results in what you have.
 
Feb 2014
54
2
England
Oh ok great, thanks! Then the question asks me to find the affine transformation g o f in the same form as I found g in part (a).
I wrote if f(X) = BX+b and g(X) = CX+c then
g o f = g(fX) = C(fX)+c = C(BX+b)+c = CBX+Cb+c = (CB)X+(Cb+c)

Is this the right formula to use?
 

Deveno

MHF Hall of Honor
Mar 2011
3,546
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Tejas
Oh ok great, thanks! Then the question asks me to find the affine transformation g o f in the same form as I found g in part (a).
I wrote if f(X) = BX+b and g(X) = CX+c then
g o f = g(fX) = C(fX)+c = C(BX+b)+c = CBX+Cb+c = (CB)X+(Cb+c)

Is this the right formula to use?
That looks correct. The "matrix" part will be CB, and the "translation part" will be translation by Cb + c, where b is the translation part of f and c is the translation part of g, and C is the matrix associated with g.
 
Feb 2014
54
2
England
That's great, thank you!
So I now have the affine transformation g o f which I found to be (0,1,-1,0)X + (4,0).
The last part of the question asks
Use your affine transformation g o f to show that there is exactly one point
(x, y) such that the image of (x, y) under g o f is (x, y). State the
coordinates of this point.
If you could please just help me with this last part I would be so grateful :)
 

romsek

MHF Helper
Nov 2013
6,666
3,003
California
That's great, thank you!
So I now have the affine transformation g o f which I found to be (0,1,-1,0)X + (4,0).
The last part of the question asks
Use your affine transformation g o f to show that there is exactly one point
(x, y) such that the image of (x, y) under g o f is (x, y). State the
coordinates of this point.
If you could please just help me with this last part I would be so grateful :)
you can manage this. Just solve for

$v=Av+b$ where

$v=\left(\begin{array}{r}x \\ y\end{array}\right)$, $A=\left(\begin{array}{rr} 0 &1 \\ -1 &0 \end{array}\right)$, and $b=\left(\begin{array}{r} 4\\ 0 \end{array}\right)$
 
Feb 2014
54
2
England
Thank you so much for all your help on this question. I actually understood it once you explained, just think I needed that initial start. My final answer is x=2 and y=-2, which, when I draw the diagram, seems to be correct. Thanks once again :)