Page 229 - Advanced Linear Algebra
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Real and Complex Inner Product Spaces 213
set is orthonormal if
º" Á " » ~ Á
for all Á 2 , where Á is the Kronecker delta function.
"
Of course, given any nonzero vector #= , we may obtain a unit vector by
multiplying by the reciprocal of its norm:
#
"~ #
# ))
This process is referred to as normalizing the vector . Thus, it is a simple
#
matter to construct an orthonormal set from an orthogonal set of nonzero
vectors.
Note that if "# , then
) )"b# ) ) ~ " ) ) b #
and the converse holds if -~ s .
Orthogonality is stronger than linear independence.
Theorem 9.9 Any orthogonal set of nonzero vectors in = is linearly
independent.
Proof. If E ~¸" 2¹ is an orthogonal set of nonzero vectors and
" bÄb " ~
then
~ º " b Ä b " Á " » ~ º"Á "»
E
and so ~ , for all . Hence, is linearly independent.
Gram–Schmidt Orthogonalization
The Gram–Schmidt process can be used to transform a sequence of vectors into
an orthogonal sequence. We begin with the following.
Theorem 9.10 (Gram–Schmidt augmentation ) Let = be an inner product
space and let E ~¸" Á Ã Á " ¹ be an orthogonal set of vectors in = . If
# ¤ º" ÁÃÁ" », then there is a nonzero " = for which ¸" ÁÃÁ" Á"¹ is
orthogonal and
º" ÁÃÁ" Á"» ~ º" ÁÃÁ" Á#»
In particular,
