Polynomial ring/Endomorphism/Inserting/Ring homomorphism (without concept)/Fact

Let ${}K$ be a field, ${}V$ a ${}K$-vector space, and

$f\colon V\longrightarrow V$

a linear mapping. Then the mapping

$K[X]\longrightarrow \operatorname {End} _{}{\left(V\right)},P\longmapsto P(f),$

has the following properties.

1. For a constant polynomial ${}P=a_{0}$, we have

   ${}P(f)=a_{0}(f)=a_{0}f^{0}=a_{0}\operatorname {Id} _{V}\,.$

   In particular, the zero polynomial is sent to the zero mapping and the constant ${}1$-polynomial is sent to the identity.

2. We have

   ${}(P+Q)(f)=P(f)+Q(f)=Q(f)+P(f)\,$

   for all polynomials ${}P,Q\in K[X]$.

3. We have

   ${}(P\cdot Q)(f)=P(f)\circ Q(f)=Q(f)\circ P(f)\,$

   for all polynomials ${}P,Q\in K[X]$.

4. We have

   ${}(X^{n})(f)=f^{n}\,$

   for all ${}n\in \mathbb {N}$.

Proof, Write another proof