Affine transformation question...HELP!

[alexlbrown59]

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]

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)

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

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

$f(x)=r(x-t)+t$

 

[alexlbrown59]

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?!

 

[romsek]

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]

Ok I'm wrong. When you take my expression and expand it out it results in what you have.

 

[alexlbrown59]

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]

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.

 

[alexlbrown59]

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]

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 $b=\left(\begin{array}{r} 4\\ 0 \end{array}\right)$

 

[alexlbrown59]

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