Lecture Il-

Final will not include Chapter 3 .

Chapter 13 order does matter

M = PDP"on D= p

-'

MP

Chapter 14-

14.1 Properties of the standard Basis

Y; !"Ie..

. . . ,en3 ei " g)← ith positionE IR

' Ee ..es - fl 'd .fi) )Heil -- eire -- Rie J.fi?:gJ-- Ie

:÷i :"

or:"

noon.

eiei -- si to IDefinition : A basis lbs . . . ,bn3 is orthogonal if--

bi . bj = 0 if itj .

Definite : A basis {by . . .

,but is orthonormal if

bi . bj = Sij .

To find a unit vector from a vector J,

i --I11811

'

Basis spans the space and is linearly

independent. Take FERN with basis{b. , . . .

,but such that the basis is orthonormal

.

J = qts , t . . - t c.In

f. Ii = (cis ,t ' ' ' tcnbn) - bi

=5i! - - - + cibi.tt?.+c.5nIif

J - bi Ci

J -- (e.5.)I , t .

. - + (t.biz= If

,

E. Ii)bT

theoremFor an orthonormal basis {I, . . ,5n3, any

vector I can be expressed as

J =.

(J -Bibi

t l!? .LI! it's]J . I = I}] . (f) = 2i. E- [31191--3E- zfj ) -1319)# D. till limit. -in -- oI, B

.

is.

-

- filler -- f."⇒ in -- FYI.]

Hi il: I a-

i. bi -- f !Y÷) . {3) = sire tyre -- StrJ -E- f- ) . = -statue -- - tr

i -- Tal' -Hii:] -- I :]✓= Naib)

fix) -- X g Cx)=3x

-5,gEV,then f. g

-- fabflxlgcxldx

{ I , cos nx ,sin Mx) on [0,2T) ,NEZ←

["

cos nx dx = 's sin nx /!= In sin 2nF - th sin 0

= 0

So"sin n x dx = - th cos nx IF

= th cos O - th cos 2hIT

= In - th = O

J? cos nx sin mxdx = -nsihm×smxosnxµO

=

-4 inZ2nmos2nA

+nsitOm2 - n2

= 0,if m Fn

= - cos' (nx) "t⇒ it m = n

O

= -Co5(2uws2(o72h

= 0

Sit - dx = x hi ⇐

HAI -FF

J!"

cosTux) dx=Znx t sin thx) at

÷=4he + sinL4#°47

-

Orsino4h

= IT

Ji"

sin' CnAdx = it

{¥* , ⇐ cos nx ,sinmx ) in [o,zit) for n,m C-I

find =n

SITE) . + f in nxtrfinnxtfff.com#cosnx

= ¥µ Co + ansinnx tbh cos nx

co = -¥? Axl dxan = tf!

"

Xx) sin nxdx

bn = #["

Axl cos ux dx

six =#In+fun+1*-1

. .-

tix FIT

B = It , x, x', -. . }

E B = { I , 2Kx- E) ) (o, if

-5, f

[f , - f, dx = S ! l . I dx = x / o'= I

f.'

Lakx-⇒Jdx =/!Xx ' - xt Ii ) dx= if - Et Ix) ) != if's - I + I )= 12kt= /

fi fi . Edx = So"

253 (x - E) dx= 25¥ - Ix))!= ers (E - E)= 0

Definition A matrix P is ont if F'=P?

theorem A changeof basis matrix P relating two orthonormal

bases is an orthogonal matrixp-I= pt

.

I5 , in 5

,

ii.E-ottztr-fztr-obz.55-o-EI-E.lt#E=o ✓5. I,=¥ - tr -to

÷÷i÷÷ii÷÷÷±÷: :p

Ii - B;-

- Sig.

I - (sij)

14.4 Gram - Schmidt orthogonal izationf. Jt = O

" a

ftTt = J - J H

a. ft -- ii. ft - Eua)= I -J - vi.8

¥sat

= it . J - it .I

= O

J = T " + It is the ot¥¥tio of

J

J " . Jt = J " . (J - J H)= JH n

f - J " . JH

= u- o -Ea*

= O

{vi. Jt) is orthogonal

{it ,E. E )

ut-w-E.EU - it

q A

part of w part ofE

parallel tois parallel to

it

linearly independentset {vi.T.is. . . - 3. we can define an orthogonal basisfor span This , . - - 3 as

If = 8 ,

If = J -TF -VT - t

-V'

ii. is - - it

ii. a - in - it - - - - - it

span {viii.Js, . .. 3 -- span {It, it, . . . }

' ftp.l !11111¥ : ±

ii. -- ri -- f :o)E- i . - it

Titi:*,tis .

ist -- i.- it - it

-- lit ' it ::÷. Kol

-

- lit til= ft)kill :tillit til I

Qtstactonizahias