Show that translations and scalings commute among themselves but not with each other; i.e., T T' = T' T and S S' = S' S but T S and S T are generally not equal. how can i show that they are generally not equal?
Start by writing out what "translations" and "scalings" mean. If T is a translation, on $\displaystyle R^n$, say, then [tex]T(x_1, x_2, ..., x_n)= (x_1+ a_1, x_2+ a_2, ..., x_n+ a_n). A "scaling" is of the form $\displaystyle S(x_1, x_2, ..., x_n)= (ax_1, ax_2, ..., ax_n)$. Just do the calculations for TS and ST.