Chapter 6.1, Problem 76E

A translation in $R^2$ is a function of the form $T\left(x,y\right)=\left(x-h,y-k\right)$, where at least one of the constants $h$ and $k$ is nonzero.

(a) Show that a translation in $R^2$ is not a linear transformation.

(b) For the translation $T\left(x,y\right)=\left(x-2,y+1\right)$, determine the images of $\left(0,0,\right),\left(2,-1\right),$ and $\left(5,4\right)$.

(c) Show that a translation in $R^2$ has no fixed points.