What is the difference between codomain and range? My book says that if there is a linear transformation $T: V \to V'$, then $V'$ is the codomain of $T$ but it also says that $T[V]$ is the range of $T$. $T[V]$ the same as $V'$?

The codomain need not be the same as the range. Take any projection operator like $\begin{bmatrix}1&0\\0&0\end{bmatrix}$; its codomain is $\mathbb R^2$ but its range is only the subspace spanned by $(1,0)^T$.
However, it is always true that $T(V)\subseteq V'$ and that the transformation can be restricted to its range ($T': V\to T(V)$) such that range and codomain are equal.
