Prove that the transformation UT is not invertible. > Let T be a linear transformation from $R^3$ into $R^2$, and let U be a linear transformation from $R^2$ into $R^3$. Prove that the transformation UT is not invertible. Generalize the theorem. I am thinking that I must use this Corollary: > If $T \in L(V,W), U \in L(W,Z)$ invertibles both, then $UT \in L(W,Z)$ is also invertible and $(UT)^{-1} = T^{-1}U^{-1}$ Can I get some help?

Suppose $UT$ is invertible, then $UTZ = I$, where $I$ is the identity on $\mathbb R^3$. However, this implies that $U(TZ) = I$ , so that $U$ is invertible. But $U$ is not invertible, since by the rank-nullity theorem, its rank must be atmost two, hence it is not surjective.

You can see how to generalize this : see that $3 \geq 2$ played a role here. Now, how can you modify $3$ and $2$ so that the argument above still works?