Lecture 19:

Recall : n < T.fi ) I ;- -T e Spank T , .ir , . -

. sin } ) ⇒ T - - I a Tritt① - i = I

orthogonal

S'

② G - S process . , S"

{ Ii , Is . . . inn }ms { t.TT }

" orthogonalBasis , L .

I.

{ II!Yin - ni. usSpano

's -. Spanos ,

The above construction of an orthogonal basis is called

Gram - Schmidt process .

Example :

Consider V = PUR ) equipped with the inner product

< f, g 7 =L

'

,

ft gets dt .Let p= { I ,

x,

x ? - - .,

X"

, -

- - } be standard ordered basis

for PHR ) .

Take T , = I . O ," "

Then : Tz = x - CK ,T

. > -- V , = xHv.

no2

is . . i . a ;÷9%, -ax3v in = a- Iz

11 Til 3

i. as .a÷i . . '= X

'- ZX and So on i - - -

for PUR)

This produces an orthogonal basis{ Ti

,Tz

,.

.

},

whose

elements are called Legendre polynomial .

Corollary : Let V be a non - zero finite - diminner product space .

Then, V has an orthonormal

basis p = { I , , . . ,Tn3Sit .

HI EV,

wehave = I =

,

Ti > Tri

G - olla - y : Let V be a non- zero finite - dim inner product space

with an orthonormal basis p - {I

, ,Tz , . . , In } . Let T be

a linear operator on V . Let A = ET ]p . Then : Aij-LTCTjs.vn .

Proof : [ TIP =/ . . . fityjgyp.

..

)TED -- it! iTi

c- Aij

Orthogonal complement

Def : Let S be a non - empty subset of an inner product

Space V.

The orthogonal complement of S is definedas :

St Eet { Ie V . . s I , I >= o for V-J e S }

-

Proposition : Let V be an inner product space and Wc ✓ a

finite - dim subspace of V . Then i tty EV , I ! it E W and I C- Wt

such that J -- It E .

Furthermore , if {I

, .ir , . . . step} is an orthonormalbasis for W

,

hthen : I = -2 Bi

i =L

The vector view is called the orthogonal projectionof

j on W .

V

Proof : Given I e V , we set indef

¥45 , Ti > Ji EWand I ihty - I . Then : I - - It I .

a

W Tnt ? ?Now , LI

,Jj > = CJ - I , Tj > = a J , Tj > - < I,Tj >

= e5,

Tj > -

i7zcy.viysTi.vj@Ez.iEsiitis.iE.r.

o

. .

" in

.

'

. I e Wt

For uniqueness , suppose IT'

EW and I'

E Wt Such that :

I - - It I = I 't E' ⇒ I - I

'= E ' - I E Wnwt

in

a a A wta at w ht ww w

Claim : w n wt = 3 T }

P Take J e wnwt . Then : s I , To >= 0

a a

W wt⇒ in -6

This implies : it -I ' = I ' - I -_ T

⇒ I = I'

and I -_E'

.

Corollary : With notations asabove , then :

115 - Ill 7115 - Ill forVIEW

A

and equality hFIdsFffI -_ ITW

yRe-make. Orthogonal projection is the gun . gonalvector in W closest to I . proj

.

: Let I e W . Then : J - - uit I ⇒ E-

- I -8a

w"

wt

ily-5112=11uitz - IIT =L xtI,@tE >" war It "w "w Ta

= L T - I,

I - I > ten - I. I > + CE , I - I > t SEE >11 It

. il

o O

= Itu - Ill'

t HER z HEH'

= 115 - FIT

The equality holds iff Hui - Ill'

-_ o iff I -- I .

Proposition : Suppose S = { I , ,Tz , . - . , Th } is anorthonormal

pet in an n - dimensionalinner product space V . Then :

( a ) S can extended to anorthonormal basis

{ Ji , -Vz , . - , Th , Treat , , . . . ,In } for V .

(b) If W= span I S ) , then S ,= { They , . . ,Tn3 is

an orthonormal basis for Wt

(c) If W is any subspaceof V

,then :

dim I V ) = dimLw ) t dim C

Wt )

Proof : ( a ) we first extends to abasis i

a{ Ii

, . . ,

Th,

The,

-. .

,In } for V .

-

L . I .

Then, we apply the G - S process

to this basis .

'

.

'

S is orthonormal , .'

,Ti

,. .

,Tres remains the same

during the G - S process .

So,

this process gives an orthonormalbasis for V of

the form { Ti , Ja , . . , Th , The , , . . ,In }

--

unchangednew

(b) Note :. S , = { The , . . ,Tn3 is linearly independent (

'

i

'

it is

part of• S

,c Wt ( We span i , . . ,Th) )

the basis )

⇒ Spanish cwt

Suffices to show = Span C S , )= Wt

.

For Ie V,

we have I = ¥45IiIf I Ewf then C I , Ii > = o for 5=1,2 ,

- - ik

n

Then : I - I a I , Tri > I ; E Span LSDj= htt

! . Wtc Span I s , ) ⇒ w 'T Span CS , )

Kc ) For any W , choose anorthonormal basis { Ji

, . . ,Trees

for W and extendit to an orthonormal basis

{ Tl , - - , Th , Berti , - . ,En } for V .

Then :

dim LV ) = h= kt C n - h )

= dinlw ) t dim I Wt )