# To show the function is well defined

• Jan 2nd 2010, 09:38 PM
math.dj
To show the function is well defined
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..
• Jan 3rd 2010, 02:11 AM
tonio
Quote:

Originally Posted by math.dj
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..

In this case, to show it is well-defined you must show that $[v]:=v+W_i= v'+W_i=:[v']\,\Longrightarrow T([v])=T(v+W_i)=T(v'+W_i)=T([v'])$

Tonio