Just as a matrix has a rank and a nullity, so too do linear transformations. And just like the rank and nullity of a matrix are related (they sum to the number of columns, Theorem RPNC) the rank and nullity of a linear transformation are related. Here are the definitions and theorems, see the Archetypes (Archetypes) for loads of examples.
Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then the rank of $T$, $\rank{T}$, is the dimension of the range of $T$, \begin{equation*} \rank{T}=\dimension{\rng{T}} \end{equation*}
Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then the nullity of $T$, $\nullity{T}$, is the dimension of the kernel of $T$, \begin{equation*} \nullity{T}=\dimension{\krn{T}} \end{equation*}
Here are two quick theorems.
Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then the rank of $T$ is the dimension of $V$, $\rank{T}=\dimension{V}$, if and only if $T$ is surjective.
Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then the nullity of $T$ is zero, $\nullity{T}=0$, if and only if $T$ is injective.
Just as injectivity and surjectivity come together in invertible linear transformations, there is a clear relationship between rank and nullity of a linear transformation. If one is big, the other is small.
Suppose that $\ltdefn{T}{U}{V}$ is a linear transformation. Then \begin{equation*} \rank{T}+\nullity{T}=\dimension{U} \end{equation*}
Theorem RPNC said that the rank and nullity of a matrix sum to the number of columns of the matrix. This result is now an easy consequence of Theorem RPNDD when we consider the linear transformation $\ltdefn{T}{\complex{n}}{\complex{m}}$ defined with the $m\times n$ matrix $A$ by $\lt{T}{\vect{x}}=A\vect{x}$. The range and kernel of $T$ are identical to the column space and null space of the matrix $A$ (Exercise ILT.T20, Exercise SLT.T20), so the rank and nullity of the matrix $A$ are identical to the rank and nullity of the linear transformation $T$. The dimension of the domain of $T$ is the dimension of $\complex{n}$, exactly the number of columns for the matrix $A$.
This theorem can be especially useful in determining basic properties of linear transformations. For example, suppose that $\ltdefn{T}{\complex{6}}{\complex{6}}$ is a linear transformation and you are able to quickly establish that the kernel is trivial. Then $\nullity{T}=0$. First this means that $T$ is injective by Theorem NOILT. Also, Theorem RPNDD becomes \begin{equation*} 6=\dimension{\complex{6}}=\rank{T}+\nullity{T}=\rank{T}+0=\rank{T} \end{equation*} So the rank of $T$ is equal to the dimension of the codomain, and by Theorem ROSLT we know $T$ is surjective. Finally, we know $T$ is invertible by Theorem ILTIS. So from the determination that the kernel is trivial, and consideration of various dimensions, the theorems of this section allow us to conclude the existence of an inverse linear transformation for $T$. Similarly, Theorem RPNDD can be used to provide alternative proofs for Theorem ILTD, Theorem SLTD and Theorem IVSED. It would be an interesting exercise to construct these proofs.
It would be instructive to study the archetypes that are linear transformations and see how many of their properties can be deduced just from considering only the dimensions of the domain and codomain. Then add in just knowledge of either the nullity or rank, and see how much more you can learn about the linear transformation. The table preceding all of the archetypes (Archetypes) could be a good place to start this analysis.