<s>[INST] Finding the affine transformation that will change a given ellipse into the unit circle $x^2$ + $y^2$ =1. We are given that ellipse $E$ is given by $x^2+4y^2-2x+16y+1=0$ and we are asked to find $t_2 \in A(2)$ such that $t_2(E)$ is the unit circle. 
 [/INST] Transformation is - tranlsation + scaling
$$
x^2+4y^2-2x+16y+1=x^2-2x+1+4(y^2+4y+4)-16=(x-1)^2+4(y+2)^2-16=0 \\
\frac{(x-1)^2}{16}+\frac{(y+2)^2}4=1
$$
So if you change coordinates
$$
\xi = \frac{x-1}4 \\
\eta= \frac{y+2}2
$$
your ellipse will become $\xi^2+\eta^2=1$
Update
Transformation itself can be represented in vector form $\mathbf x' = A \mathbf x + \mathbf b$
$$
\left [ \begin{array}{c}
\xi \\ \eta
\end{array}\right ] = f \left ( \left [ \begin{array}{c}
x \\ y
\end{array}\right ]\right ) = \left [ \begin{array}{c}
\frac x4 - \frac 14 \\ \frac y2 + 1
\end{array}\right ] = \left [ \begin{array}{cc}
\frac 14 & 0 \\
0 & \frac 12
\end{array} \right ] \left [ \begin{array}{c}
x \\ y
\end{array}\right ] + \left [ \begin{array}{c}
-\frac 14 \\ 1
\end{array}\right ]
$$
</s>