4. Homogeneous coordinates used in computer graphics: (a) Translations (moving r units to the right and s units up): Let " E R2 and T : R2 + R2 be defined by T(u) = ut Show that if r 0 or s * 0 then T is not a linear transformation of R2. (b) Now we see that translation is not a linear transformation of R'. In order to represent translation as a linear transformation, we need to work in R3. We introduce the homogeneous coordinate system by writing each vector in R2 as the vector in R3. When we want to plot the point given in homogeneous coordinate system, we ignore the third coordinate and plot in R2. (Equivalently, we can think of drawing on the plane z = 1). For example, if we want to scale by a factor of c, we use the linear transformation Sc : R3 - R3 defined by S cy in the homogeneous coordinate system. The matrix of S. is 0 0 0 C 0 0 i. Let Ty,s : R3 -> R3 be the linear transformation defined by str Tr,s y + s Write down the matrix of Tr,s. (Note: it is a 3 x 3 matrix.) ii. Show that Tres is an isomorphism of R3