rank of a linear mapping

The rank of a linear mapping $L:U\to V$ is defined to bethe $dimL(U)$, the dimension of the mapping’s image. Speaking lessformally, the rank gives the number of independent linear constraintson $u\in U$ imposed by the equation

$$L(u)=0.$$

Properties

1. If $V$ is finite-dimensional, then $\mathrm{rank}L=dimV$ if and onlyif $L$ is surjective.

2. If $U$ is finite-dimensional, then $\mathrm{rank}L=dimU$ if and onlyif $L$ is injective.

3. Composition of linear mappings does not increase rank. If$M:V\to W$ is another linear mapping, then

   $$\mathrm{rank}ML\le \mathrm{rank}L$$

   and

   $$\mathrm{rank}ML\le \mathrm{rank}M.$$

   Equality holds in the first case ifand only if $M$ is an isomorphism, and in the second case if andonly if $L$ is an isomorphism.