Describe the image of (i) the x-axis, (ii) the unit disk x^2+y^2 ≤1, (iii) the unit square 0 ≤x,

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Describe the image of (i) the x-axis, (ii) the unit disk...

Describe the image of (i) the $x$-axis, (ii) the unit disk $x^2+y^2 \leq 1$, (iii) the unit square $0 \leq x, y \leq 1$, under the following affine transformations: (a) $T_1\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}2 \\ -1\end{array}\right)$, (b) $T_2\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{ll}3 & 0 \\ 0 & 2\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}-1 \\ 0\end{array}\right)$, (c) $T_3\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 2\end{array}\right)$, (d) $T_4\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 0\end{array}\right)$, (e) $T_5\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}.6 & .8 \\ -.8 & .6\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}-3 \\ 2\end{array}\right)$, (f) $T_6\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{cc}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 0\end{array}\right)$. (g) $T_7\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}1 & 1 \\ -1 & 1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}2 \\ -3\end{array}\right)$, (h) $T_8\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}2 & 1 \\ -2 & -1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 1\end{array}\right)$.

Describe the image of (i) the $x$-axis, (ii) the unit disk $x^2+y^2 \leq 1$, (iii) the unit square $0 \leq x, y \leq 1$, under the following affine transformations:
(a) $T_1\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}2 \\ -1\end{array}\right)$,
(b) $T_2\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{ll}3 & 0 \\ 0 & 2\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}-1 \\ 0\end{array}\right)$,
(c) $T_3\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 2\end{array}\right)$,
(d) $T_4\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 0\end{array}\right)$,
(e) $T_5\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}.6 & .8 \\ -.8 & .6\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}-3 \\ 2\end{array}\right)$,
(f) $T_6\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{cc}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 0\end{array}\right)$.
(g) $T_7\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}1 & 1 \\ -1 & 1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}2 \\ -3\end{array}\right)$,
(h) $T_8\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}2 & 1 \\ -2 & -1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 1\end{array}\right)$.

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