Page 468 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Thus

which proves that v is orthogonal to every vector in the row space of A.

Proof (b) Since the column space of A is the row space of (except for a difference in notation), the proof follows by
applying the result in part (a) to .

The following example shows how Theorem 6.2.6 can be used to find a basis for the orthogonal complement of a subspace of
Euclidean n-space.

EXAMPLE 6 Basis for an Orthogonal Complement
Let W be the subspace of spanned by the vectors

Find a basis for the orthogonal complement of W.

Solution

The space W spanned by , , , and is the same as the row space of the matrix

and by part (a) of Theorem 6.2.6, the nullspace of A is the orthogonal complement of A. In Example 4 of Section 5.5 we
showed that

form a basis for this nullspace. Expressing these vectors in the same notation as , , , and , we conclude that the
vectors
form a basis for the orthogonal complement of W. As a check, the reader may want to verify that and are orthogonal to
, , , and by calculating the necessary dot products.

Summary

We leave it for the reader to show that in any inner product space V, the zero space {0} and the entire space V are orthogonal