Linear Transformation Equality

[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) 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

 

[lanedance]

how about something like this as a strawman:

first recognise that range is given by the span of the column vectors

now consider T*u, for an arbitrary vector u. The results is a linear combination of the vectors in T*

So the columns of T*T can be thought of as linear combinations of the columns of T*

Then you probably need to invoke that the dimension of the column and row space are the same to finish

 

[jnava]

I am slowly pounding through it. This is what I have:

Let w exist in R(T*T), then there exists a v in V s.t.

T*T(v) = w
=> T(v) exists in R(T*) => w exists in R(T*) => R(T*T) is a subset of R(T*)

The other way is trickier