Page 486 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The preceding step-by-step construction for converting an arbitrary basis into an orthogonal basis is called the
Gram–Schmidt process.

EXAMPLE 7 Using the Gram–Schmidt Process

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

vectors , , into an orthogonal basis ; then normalize the orthogonal

basis vectors to obtain an orthonormal basis .

Solution

Step 1.

Step 2.

Step 3.

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