What is the composition of T and T in terms of linear transformations?

[BenZino11]

Let u (not equal to 0) be a vector in R^2 and let
T: v --> proju(v)

1. Show that T is a linear transformation.
2. Describe the composition T  T.
3. If ~u = [1,−1], find the standard matrix for T.

I'm good with 1 and 3, but I'm not sure what 2 is asking. Excuse the poor notation, it's my first time using this site.

 

[Office_Shredder]

What are the little boxes supposed to be?

 

[BenZino11]

Sorry, I don't see little boxes. Maybe a computer issue but its just:

ToT (I'm not sure if this is T dot T or if the larger circle represents something else)

Thanks!

 

[Mark44]

I see three boxes: T  T. It's probably a thing with the browser, IE 8.0 seems to be
unable to display certain characters.

If T(v) is the project of v onto u, what happens if you take the projection of the projection of v? I.e., what is T(T(v))?

 

[BenZino11]

( (u.[( (u.v)/(u.u) )u])/u.u ) u

Is that all?

 

[Mark44]

You're missing my point. Once you project the vector for the first time, you have already "flattened" it out in the direction of u. What will happen if you try to flatten it out again?

 

[BenZino11]

The length will remain the same, as the projection of the vector v on u is the vector itself.
So I suppose to describe the composition I would just write out the original projection formula?