Orthogonal Complements Prove the properties stated in Problem using the following definiti...

Orthogonal Complements Prove the properties stated in Problem using the following definition, illustrated by Fig. 4. Assume that is a subspace of ℝn.

Orthogonal Complement

Let be a subspace of ℝn. A vector is orthogonal to subspace provided that is orthogonal to every vector in . The set of all vectors in ℝn that are orthogonal to is called the orthogonal complement of , denoted

.

Figure 4 An orthogonal complement to a plane