First, some clarifications. We are considering linear maps from a finite-dimensional vector space to itself (also called an operator L(V)), as opposed to linear maps from one vector space to another vector space.

Suppose V is a finite-dimensional vector space, TL(V), and λ ∈ F.

Some definitions to know:



         <=> V has a basis consisting of eigenvectors of T

         <=> ∃ 1-dimensional subspaces U1, …, Un of V, each invariant under T, such that V          = U1 ⊕ … ⊕ Un

         <=> V = E(λ1, T) ⊕ … ⊕ E(λm, T)

         <=> dim V = dim E(λ1, T) + … + dim E(λm, T)