Page 464 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
P. 464
DEFINITION
Two vectors u and v in an inner product space are called orthogonal if .
Observe that in the special case where is the Euclidean inner product on , this definition reduces to the
definition of orthogonality in Euclidean n-space given in Section 4.1. We also emphasize that orthogonality depends on the
inner product; two vectors can be orthogonal with respect to one inner product but not another.
EXAMPLE 3 Orthogonal Vectors in
If has the inner product of Example 7 in the preceding section, then the matrices
are orthogonal, since
Calculus Required
EXAMPLE 4 Orthogonal Vectors in
Let have the inner product
and let and . Then
Because , the vectors and are orthogonal relative to the given inner product.
In Section 4.1 we proved the Theorem of Pythagoras for vectors in Euclidean n-space. The following theorem extends this
result to vectors in any inner product space.
THEOREM 6.2.4
