Suppose T:V-->V is a nilpotent linear map.Then there exists subspaces W1,W2,...Wk of V such that

1)T(Wi) is contained in Wi ,for i=1,...,k

2)Each Wi is T-cyclic subspace of V

3)V=direct sum of Wi's ,for i=1,...k.

In the proof of this question ,as T(Wi) is contained in Wi,we get a induced linear map,T':V/Wi-->V/Wi defined by T'([v])=[Tv],for all v in V.

how do i show that the map is well defined..