Adjoint representation (faithfulness on semisimple Lie algebras)

Let $L$ be a finite dimensional Lie algebra and $Z(L)$ the center of $L$. Show that $\operatorname {ad}: L \to \mathfrak {gl}(L)$ is a homomorphism of the Lie algebra $L$ with kernel $Z(L)$.

Show that the adjoint representation of a semisimple Lie algebra is faithful. A faithful representation of a Lie algebra is a linear representation in which different elements of $L$ are represented by distinct linear mappings.