Page 224 - Advanced Linear Algebra
P. 224
208 Advanced Linear Algebra
º%Á &» c º&Á %» ~
These two equations imply that º %Á&» ~ for all %Á& = and so part 1)
implies that ~ . For the last statement, rotation by degrees in the real
plane s has the property that º#Á #» ~ for all . #
Norm and Distance
If is an inner product space, the norm , or length of # = is defined by
=
))#~ j º#Á #» (9.1 )
A vector is a unit vector if # )) ~ . Here are the basic properties of the norm.
#
Theorem 9.3
)
1 ))# and ))#~ if and only if # ~ .
)
2 For all - and # = ,
#
#
)) ~ ( () )
3 )(The Cauchy–Schwarz inequality ) For all "Á # = ,
( (º"Á #» " ) ) # )
)
with equality if and only if one of and is a scalar multiple of the other.
"
#
4 )(The triangle inequality ) For all "Á # = ,
) )"b# ) ) )" b # )
with equality if and only if one of and is a scalar multiple of the other.
#
"
)
5 For all "Á #Á % = ,
) )"c# ) )"c% b ) )%c#
)
6 For all "Á # = ,
( "c# ()) ) ) ) " c #
)
7 )(The parallelogram law ) For all "Á # = ,
) )"b# b ) )"c# ) ~ " ) ) b # )
Proof. We prove only Cauchy–Schwarz and the triangle inequality. For
Cauchy–Schwarz, if either or is zero the result follows, so assume that
"
#
"Á # £ . Then, for any scalar -,
" c # )
)
~º" c #Á " c #»
~ º"Á "» c º"Á #» c ´º#Á "» c º#Á #»µ
Choosing ~ º#Á "»°º#Á #» makes the value in the square brackets equal to
