rank of a linear mapping

The rank of a linear mapping $L\colon U\to V$ is defined to be the $\dim L(U)$ , the dimension of the mapping's image. Speaking less formally, the rank gives the number of independent linear constraints on $u\in U$ imposed by the equation$$ L(u)=0.$$

Properties

1. If $V$ is finite-dimensional, then $\rank L=\dim V$ if and only if $L$ is surjective.
2. If $U$ is finite-dimensional, then $\rank L=\dim U$ if and only if $L$ is injective.
3. Composition of linear mappings does not increase rank. If $M\colon V\to W$ is another linear mapping, then$$\rank ML \le \rank $$ and$$\rank ML \le \rank M$$ Equality holds in the first case if and only if $M$ is an isomorphism, and in the second case if and only if $L$ is an isomorphism.