# Thread: Linear Transformations and Range

1. ## Linear Transformations and Range

Hi, I am trying to prove the following equality

Range(T*T) = Range(T*)
where T is a linear transformation and * denotes the adjoint.

I know I must first show that Range(T*T) $\displaystyle \subset$ Range(T*) and vice versa.

so, Let w exist in R(T*T), then there exists a v in vector space V s.t.
T*T(v) = w.
Then I draw a blank, what's next??? Or am I even starting it correctly?

Thanks

2. ## Re: Linear Transformations and Range

Originally Posted by jnava
Hi, I am trying to prove the following equality

Range(T*T) = Range(T*)
where T is a linear transformation and * denotes the adjoint.

I know I must first show that Range(T*T) $\displaystyle \subset$ Range(T*) and vice versa.

so, Let w exist in R(T*T), then there exists a v in vector space V s.t.
T*T(v) = w.
Then I draw a blank, what's next??? Or am I even starting it correctly?

Thanks

3. ## Re: Linear Transformations and Range

Originally Posted by Drexel28
Because I can just use T(v) as a vector all on its own, so then T*(v) = w which implies that w is in the R(T*)

Am I right?

4. ## Re: Linear Transformations and Range

Originally Posted by jnava
Because I can just use T(v) as a vector all on its own, so then T*(v) = w which implies that w is in the R(T*)

Am I right?
Yes! So how about the second part? It's slightly trickier.

5. ## Re: Linear Transformations and Range

Do I just use the same approach?

Let w exist in R(T*), then there exists a v in vector space V s.t.
T*(v) = w. Then I am guessing there is an adjoint property to get to T*T(v) = w.....??