Question

3. Let P be an nxn projection matrix (which means that it satisfies P²=P and P^T=P). The goal of this problem is to show that P as a linear transformation is the orthogonal projection onto the range of P.

(a) Show that if v is in the range of P, then Pv=v.

(b) Show that if v is orthogonal to the range of P then Pv=0. (Hint: PPv lies in the range of P, hence is orthogonal to v. Study the dot product of v with PPv.)

(c) The previous parts show that P and the projection map act the same way on vectors that lie either in the range of P or the orthogonal complement of it. Why does this imply that P and the projection map agree on all vectors?

Answer 1

a) Let , then such that

                                              

b) Let be orthogonal to the Range(P), then: <.> (denotes dot product)

                                    

For each . This we already proved.

Then,

Now, in particular :

                            

Rewriting the dot product:

So,

This is only possible if .

This completes the proof.

c) If two linear transformations map a basis to the same image then they are equal!!

We can write any vector as a sum of a vector in Range(P) and in the Nullspace of P. This is true from the Rank-Nullity theorem.

                                Orthogonal complement of Range(P)

The projection map maps this to w. And:

                           

Hence, the projection matrix and map agree on all vectors.