Orthogonal Decomposition

I proceed to define the orthogonal decomposition for some vector , where , where is a subspace of , where is the Orthogonal Complement of , where is the orthogonal projection of onto , and is a vector orthogonal to . This is a Vector Decomposition.

• Every has a unique sum in the form above, so long as is a subspace of

Concerning

If is an Orthogonal Basis for , then , the orthogonal projection of onto is given by:

See Best Approximation, but in essence is the closest vector in to

Examples:

HW 6.3 Q1 HW 6.3 Q5 HW 6.3 Q6