Proving Cyclic Decompositions: Let T be a Linear Operator on V

[Bhatia]

Let T be a linear operator on the the finite dimensional space V, and let R be the range of T.

(a) Prove that R has a complementary T-invariant subspace iff R is independent of the null space N of T.

(b) If R and N are independent, prove that, N is the unique T-invariant subspace complementary to R.

I supposed R has a complementary T - invariant subspace, say, W. Then , R should be T- admissible. I assumed to the contrary, that R intersection T is not equal to {0}. I took a point in the intersection but could not proceed further. Please suggest.

 

[arkajad]

Can you define "T-admisible"?

 

[Bhatia]

Can you define "T-admisible"?

Thanks for your reply.

Given a linear operator T on a vector space V then a subspace W is T- admissible if

i) W is invariant under T
(ii) if f(T) β belongs to W, there exists a vector η in W such that f(T)β = f(T) η.

 

[arkajad]

What is f(T) in (ii)?

 

[morphism]

Hint: Fix a basis for the range of T and extend it to a basis for V. What can you say about these extra basis vectors?