Let $V$ be a finite-dimensional vector space and $T:V\to W$ a linear transformation. Take $\{v_1,\dots,v_r\}$ to be a basis for $\ker T$, and complete this set into a basis $\{v_1,\dots,v_r,u_1,\dots,u_m\}$ of $V$. Taking the spans of $v_1,\dots,v_r$ and $u_1,\dots,u_m$ separately, we obtain $V=U_1\oplus U_2$ where $U_1$ is the kernel of $T$ and $U_2$. It's easy to see that the image of $T$ is isomorphic to $U_2$.
The question: is there a "nice" characterization of $U_2$ which does not involve completing the basis of the kernel? The kernel itself is defined in a very simple way: $\ker T=\{v\in V:T(v)=0\}$. Is there something similar for $U_2$?