vector space V, T in set of lin. transforms V->V. Prove....

[x3non25]

Let V be a vector space, and let T be in the set L(V), the space of linear transforms from V to V. Prove that:

1) T^2 = To if and only if R(T) is a subset of N(T), and

2) if rank(T) = rank(T^2) then R(T) intersection N(T) ={0} (hint: first prove that N(T)=N(T^2))

any help would be greatly appreciated

 

[pka]

Let V be a vector space, and let T be in the set L(V), the space of linear transforms from V to V. Prove that:
1) T^2 = To if and only if R(T) is a subset of N(T), and

2) if rank(T) = rank(T^2) then R(T) intersection N(T) ={0} (hint: first prove that N(T)=N(T^2))

These are purely ‘definitional proofs’. We apply the definitions.
\(\displaystyle T^2(v)=T(T(v))\) for all v. \(\displaystyle T(v) \in R[T]\) so if \(\displaystyle T^2=T_0\) then by definition range of \(\displaystyle T\) is a subset of the null space of \(\displaystyle T\). The other way is trivial.

For number two, what do you know about rank?
How is rank related to dim(N(T))?