Constructing a projector onto a given subspace
Let be a subspace of Let and fix some basis in Define the matrix of size (the vectors are written as column vectors).
Exercise 1. a) With the above notation, the matrix exists. b) The matrix exists. c) is a projector.
b) To see that exists just count the dimensions.
c) Let's prove that is a projector. (1) allows us to make the proof compact. is idempotent:
Exercise 2. projects onto :
Proof. First we show that Put
for any Then
This shows that
Let's prove the opposite inclusion. Any element of is of form
with some Plugging (4) in (3) we have which shows that
Exercise 3. Let be a subspace of and let . Then .
Proof. Since any element of is orthogonal to any element of , we have only to show that any can be represented as with . Let be the projector from Exercise 1, where . Put , . is obvious. For any , , so is orthogonal to . By definition of , we have .