If is a projection, then its image and the kernel are complementary subspaces, namely
Proof. Suppose that is a projection. Let be given, and set
The projection condition (1) then implies that , and we can write as the sum of an image and kernel vectors:
Specializing somewhat, suppose that the ground field is or and that is equipped with a positive-definite inner product. In this setting we call an endomorphism an orthogonal projection if it is self-dual
The kernel and image of an orthogonal projection are orthogonal subspaces.
Proof. Let and be given. Since is self-dual we have
Thus we see that a orthogonal projection projects a onto in an orthogonal fashion, i.e.
for all .
|Date of creation||2013-03-22 12:52:13|
|Last modified on||2013-03-22 12:52:13|
|Last modified by||rmilson (146)|