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
is a subspace of ℝn.
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