Page 816 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The definitions of such terms as orthogonal vectors, orthogonal set, orthonormal set, and orthonormal basis carry over to
complex inner product spaces without change. Moreover, Theorems 6.2.4, 6.3.1, 6.3.3, 6.3.4, 6.3.5, 6.3.6, and 6.5.1 remain
valid in complex inner product spaces, and the Gram–Schmidt process can be used to convert an arbitrary basis for a complex
inner product space into an orthonormal basis.

EXAMPLE 6 Orthogonal Vectors in
The vectors

in are orthogonal with respect to the Euclidean inner product, since

EXAMPLE 7 Constructing an Orthonormal Basis for

Consider the vector space with the Euclidean inner product. Apply the Gram–Schmidt process to transform the basis

vectors , , into an orthonormal basis.

Solution

Step 1.

Step 2.

Step 3.

Thus
form an orthogonal basis for . The norms of these vectors are