Prove that a linear operator is indecomposable

[toni07]

Homework Statement

Let V be a finite-dimensional vector space over F, and let T : V -> V be a linear operator. Prove that T is indecomposable if and only if there is a unique maximal T-invariant proper subspace of V.

Homework Equations

The Attempt at a Solution

I tried using the definition of decomposable with respect to matrices, but I can't manipulate it to answer this question.

 

[R136a1]

Start by providing us the definition of "indecomposable operator" and "T-maximal".

 

[toni07]

A n×n matrix A is decomposable if there exists a nonempty proper subset I⊆{1,2,...,n} such that aij=0 whenever i∈I and j∉I.

I only know the definition of maximal vector which is: A vector z such that the minimal polynomial of the operator T with respect to z = the minimal polynomial of the operator T, is called a maximal vector.