Page 230 - Advanced Linear Algebra

214 Advanced Linear Algebra

"~# c "

~
where

" ~ if
~ H º#Á" » if "£

º" Á" »
Proof. We simply set

" ~ #c " cÄc "

for all , that is,

and force "ž"

~ º"Á" » ~ º# c " c Ä c " Á" » ~ º#Á" » c º" Á" »

Thus, if "~ , take ~ and if "£ , take

º#Á " »
~

º" Á " »
The Gram–Schmidt augmentation is traditionally applied to a sequence of
linearly independent vectors, but it also applies to any sequence of vectors.
Theorem 9.11 ( The Gram–Schmidt orthogonalization process) Let
8 ~ ²#Á #Á Ã ³ be a sequence of vectors in an inner product space . Define a
=

sequence E ~ ²"Á "Á Ã ³ by repeated Gram–Schmidt augmentation, that is,

c
"~ # c " Á

~
and
where "~ #
" ~ if
~ Á H º# Á" » if "£
º" Á" »
Then is an orthogonal sequence in with the property that
E
=
º" ÁÃÁ" » ~ º# ÁÃÁ# »

for all € . Also, " ~ if and only if #  º# ÁÃÁ# c . »

Proof. The result holds for ~ . Assume it holds for c . If
», then
#  º# ÁÃÁ# c

»

#  º# ÁÃÁ# c » ~ º" ÁÃÁ" c
Writing