find :h÷!i÷dhemminger/teaching/33af18...2) among all unit vectors in rn find the one for which the...
TRANSCRIPT
1) Find the orthogonal projection of e,
an to
* seat!:H÷!i÷:BSolution : We know how to compute this projection if we
have an orthonormal basis for V. Observe that vi. Va
,
and v, are pairwise orthogonal ,
so
ur-E.nu .
-
-
,
ui-nt.nu=/ ÷÷,
and
us
Titus-
- f
farm an orthonormal basis for V.
Then we can compute that
the projection of e,
onto V is
( eiu.lu ,t le
,.ua ) Ua t Ce
,
- us )U ,=
2) Among all unit vectors in Rn,
find the one for which
the sumat all components is maximal .
Solution: The word " maximal"
hints that an inequality might
be useful,
and we've
onlyseen one inequality in this
class : the Cauchy- Schwartz inequality
saysthat
Ix - yl Ellxll 11711
for x. YER? What should we use for x and yin
this case ? We're interested in using the data of a unit
vector in R" and the sum
of its components,
solet
x=ke a unit vector in IR" and Y = ei =
( !
;) so that
)
× .
y =
,
x ; = the sum at the
components at ×
The Cauchy - Scwhartz inequality says that
I E.xiflx.gl Ellxllllyll= C D ( rn ) at )
So the gum of the components af anyunit vector is at most
Tn. Equality in CH) is achieved exactly when x is parallel
to y ,i.e
.
when × =
).
so
(is the unit
vector for which the sumat all components is maximal .
3) Find an orthonormal basis for
v -
- many! ), .
3
: We 'll use the Gram - Schmidt algorithm .
Write
solution
..
.
. µ , v. =p, )
u.
-
- (7)
u .
;÷nu=f÷÷)
vs'
= vz - v ! = vs - d,
- Va ) u,
= (I
us =¥, rat = ) r,
- a t zu .
I2
Vzt
= Vz - Vz"
= Vz - ( U ,- Vz ) U
,- ( us - b) uz =
" ¥ ." .
=
→ { f,
¥,
),
÷±)} is anorthonormal basis for V
.