Linear Transformation Equalities

[jnava]

Homework Statement

I need to show the following equalities, where T is a linear transformation and * is the adjoint.

Null(T*T) = Null(T)
Range(T*T) = Range(T)



The attempt at a solution

I know I have to show that Null(T) is a subset of Null(T*T) and then Null(T*T) is a subset of Null T...

Let v exist in Nullspace of T. Then T(v) = 0
Now let v exist in the Nullspace of T*T. Then T*T(v) = 0

Now i am lost, any help would be great. Thanks

 

[micromass]

OK, so let v in the nullspace of T. Then T(v)=0. Can you show that T*T(0)=0??

The reverse is not so easy. But take T*T(v)=0. Then <T*T(v),v>=0. Now use the property of the adjoint.

 

[jnava]

Thank you!