Polynomial ring/Endomorphism/Matrix/Insertion/Remark

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

$f\colon V\longrightarrow V$

a linear mapping. Let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$, and let ${}M$ denote the corresponding matrix. Due to fact, we have a correspondence between compositions of linear mappings and matrix multiplication. In particular, ${}f^{n}$ corresponds with ${}M^{n}$. In the same way, the scalar multiplication and the addition on the space of endomorphisms ${}\operatorname {End} _{}{\left(V\right)}$ and on the space of matrices correspond to each other. Therefore, instead of the assignment ${}P\mapsto P(f)$, we can also work with the assignment ${}P\mapsto P(M)$.