1. ## Affine Transformations

In this question, f and g are both affine transformations. The transformations f is a reflection in the line y = -x +1, and g maps the points (0,0), (1,0) and (0,1) to the points (1,5), (1,-4) and (0,-5) respectively.

a) Determine g in the form g(x) = Ax + a, where A is a 2 x 2 matrix and a is a vector with two components.

b) Write down the matrix that represents reflection in an appropriate line through the origin, and find f (in the same form as for g in part (a)) by first translating an appropriate point to the origin.

c) Find the affine transformation g o f (in the same form as for g and f in parts (a) and (b)).

d) Hence or otherwise, find the images of the points (0,0), (0,-2) (2,-2) and (2,0) under g o f. Mark these points and images on the same diagram, making it clear which points maps to which. Describe g o f geometrically as a single transformation.

In this question, f and g are both affine transformations. The transformations f is a reflection in the line y = -x +1, and g maps the points (0,0), (1,0) and (0,1) to the points (1,5), (1,-4) and (0,-5) respectively.

a) Determine g in the form g(x) = Ax + a, where A is a 2 x 2 matrix and a is a vector with two components.
First we observe that:

$\displaystyle g([0,0]')=A{0 \brack 0}+a={1 \brack 5}$

which implies that:

$\displaystyle a = {1 \brack 5}\ \ \ \dots(1)$.

Now: $\displaystyle g([1,0]')=A{1 \brack 0}+{1 \brack 5}={1 \brack -4}$, so:

$\displaystyle A{1 \brack 0}={0 \brack -9}\ \ \ \dots(2)$.

Also: $\displaystyle g([0,1]')=A{0 \brack 1}+{1 \brack 5}={0 \brack -5}$, so:

$\displaystyle A{0 \brack 1}={-1 \brack -10}\ \ \ \dots(3)$.

Now $\displaystyle (2)$ and $\displaystyle (3)$ imply that:

$\displaystyle A=\left[ \begin{array}{cc} 0&-1\\ -9&-10 \end{array}\right]$,

so:

$\displaystyle g(x)=\left[ \begin{array}{cc} 0&-1\\ -9&-10 \end{array}\right] x+{1 \brack 5}$.

RonL