Matrix of orthogonal projection

[morsel]

Homework Statement

Let A be the matrix of an orthogonal projection. Find A^2 in two ways:
a. Geometrically. (consider what happens when you apply an orthogonal projection twice)
b. By computation, using the formula:
matrix of orthogonal projection onto V = QQ^T, where Q = [u1 ... um]

Homework Equations

The Attempt at a Solution

I have no idea how to approach (a).
(b). A^2 = Q Q^T Q Q^T = Q^2 (Q^T)^2

Thanks in advance.

 

[I like Serena]

Let's start with (a).
Suppose we have 2 dimensions and A defines an orthogonal projection.
This means that any point v is projected orthogonally on a line.
What happens to this projection if we project it again on the line?

 

[morsel]

Does it go back to the original point?

 

[I like Serena]

Does it go back to the original point?

We can find on wikipedia a page http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29" :

The transformation P is the orthogonal projection onto the line m.

and:

"In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. "