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Cyclic decompostions 
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#1
Feb2712, 08:53 AM

P: 11

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 Tinvariant 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 Tinvariant 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. 


#2
Feb2712, 11:15 AM

P: 1,411

Can you define "Tadmisible"?



#3
Feb2712, 11:43 AM

P: 11

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. 


#4
Feb2712, 12:55 PM

P: 1,411

Cyclic decompostions
What is f(T) in (ii)?



#5
Feb2812, 04:36 PM

Sci Advisor
HW Helper
P: 2,020

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?



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