Define the linear transformation. Decide which of the mappings of $\mathbb R^2$ to itself given below are linear.
$$\begin{align}T_1(x,y)&=(x+2y,y-2x)&T_2(x,y)&=(x,2x+y)\\T_3(x,y)&=(1-y,x+2)&T_4(x,y)&=(2x-y,-4x+2y)\end{align}$$
Answer the following questions only for the mappings that are linear:
(a) Write each map in matrix form.
(b) Determine the image under $T_j$ of the basis vectors $e_1,e_2$. Plot roughly the image set $T(U)$ of the unit square $U=\{te_1+ue_2|0\le t,u\le1\}\subset\mathbb R^2$ in your answer book.
With regard to part B, I understand how to find the image under $T_j$ but I don't understand how to plot the image set $T(U)$ of the unit square $U=\{te_1 + ue_2 | 0 \le t, u\le 1 \}$.
Since linear maps preserve convexity and the unit square is convex, if you find the images of the vertexes of the unit square and plot their convex hull you should have what you are looking for.