Definition: Orthogonality: Let
. A vector is called orthogonal to if . For all
- This is saying no matter how we scale the vectors, they stay orthogonal to each other
- This applied the property of bilinearity that the dot product was shown to have
Definition: Let
be non-empty. Define, S-perp as
- This can be show to be a linear subspace.
- This means that means that any linear combination of orthogonal vectors will also be orthogonal
Proposition: Let
. Then, is a linear subspace of . Proof: Pick . Then, and , for all , for all , for all . Hence, is a a linear subspace.
Definition: Let
be a linear subspace of . Then, is called the orthogonal complement of . From this, this means that and that
Remark: Let
and be the collection of all vectors orthogonal to the elements of . Then the linear span of the set is the same as set: .
Pythagorean identity: let
be such that , whenever . Then, Proof: Base case: for . . Since we know that the dot product between the two elements must be zero, we can eliminate . This proves Assume, for , , whenever for , suppose, bet such that . Say that By the inductive assumption, and hence