Direct sums of subspaces
The definition of an orthogonal sum requires two things: 1) every element can be decomposed as with and 2) every element of is orthogonal to every element of Orthogonality of to implies which, in turn, guarantees uniqueness of the representation If we drop the orthogonality requirement but retain 1) and we get the definition of a direct sum.
Definition. Let be two subspaces such that The set is called a direct sum of and denoted . The condition provides uniqueness of the representation
Exercise 1. Let If is decomposed as with define Then is linear and satisfies .
Under conditions of Exercise 1, is an oblique projector of onto parallel to
Exercise 2. Prove that dimension additivity extends to direct sums: if then
Exercise 3. Let be two subspaces of and suppose Then to have it is sufficient to check that in which case
Proof. Denote By Exercise 2, If is not empty, then we can complete a basis in with a nonzero vector from to see that which is impossible.