Affine transformation question...HELP!

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 :)*

Re: Affine transformation question...HELP!

Quote:

Originally Posted by

**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).

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)

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

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

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

Re: Affine transformation question...HELP!

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?!

Re: Affine transformation question...HELP!

Quote:

Originally Posted by

**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?!

I posted the correct form for f(x). What you have is incorrect. You neglect the first translation.

Re: Affine transformation question...HELP!

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

Re: Affine transformation question...HELP!

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?

Re: Affine transformation question...HELP!

Quote:

Originally Posted by

**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?

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.

Re: Affine transformation question...HELP!

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 :)

Re: Affine transformation question...HELP!

Quote:

Originally Posted by

**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 :)

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

Re: Affine transformation question...HELP!

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 :)