homogeneous linear problem

Let $L:U\to V$ be a linear mapping. A linear equationis called homogeneous if it hasthe form

$$L(u)=0,u\in U.$$

A homogeneous linear problem always has atrivial solution, namely $u=0$. The key issue in homogeneous problemsis, therefore, the question of the existence of non-trivial solutions,i.e. whether or not the kernel of $L$ is trivial, or equivalently,whether or not $L$ is injective.