Proposition 2

Let $u$ be an endomorphism of a vector space $E$ over a commutative field $K$; for each scalar $a$, let $V_a$ be the eigenspace corresponding to $a$. Then the subspaces $V_a$ are closed under $u$ and the sum of the $V_a$ is direct.

The first assertion is clear. By definition the subspace $V_a$ is annihilated by the element $X - a$ of $K[X]$; the $X - a, a \in K$, are irreducible and pairwise non-associate; the second assertion thus follows from VII, p. 8, Th. 1.