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 because we are dealing with a basis. This fact and the general equation imply Hence for any given there exists such that Then (2) is true and, as above, We have proved