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

.