Chapter 6 - Linear Transformations - 6.4 Additional Properties of Linear Transformation - True-False Review - Page 416: h

True

Since $Rng(T)$ is four-dimensional $\rightarrow \dim [Rng(T)]=\dim P_3(R)=4$ Apply Rank Nullity Theorem: $\dim [Ker (T)]+\dim [Rng(T)]=\dim P_3R)\\ \dim [Ker(T)]=4-4=0\\ \rightarrow Ker (T)=\{0\}$ $T$ is one-to-one. Obtain: $\dim M_{23}(R)=6\\ \dim P_3(4)=4 \\ \rightarrow \dim M_{23}(R)\gt \dim P_3(4)$ Thus, $T$ is not onto.