Today §

6.4: Orthogonal izatien

Next : § 7. I : The Spectral Theorem

Reminders lease fill out your CAPES.

My Math Lab Homework # 8 : Due THURSDAY by ltispm

MATLAB QUIZ : Tomorrow OR TODAY !

FINAL EXAM : This Saturday ,11:30am - 2:30pm

Seating 112am Assignment on Triton Ed

Given an ortho normal basis { It,

Yi,

...

,lip} for a subspaceVHN ,

the orthogonal projection of a vector I er" into V is

Pnoj ✓(B) = kin

,)uI + ⇐Adult . - . tlxupyp

= UUT ± where ltlu , Is . -. Ap ]

This B the

uniquevector yet such that y . x. e Vt

.

Geometrically ,it B the closest point in V to x .

⇐ statementEmanated:Yµh9¥ns%!⇐H.

g. Projvht beat+Hiatus = few + IN§ . y , = (1) 121 tells ) t (3) t D= 9 114112=30§ . ys= (1) th # 1) + ( 3) 11 ) =3 114414 6

6.41 What if you aren't given an orthogonal basis ?

Can you even be

surethat one exists ?

Ea. v= Null it} 't →liiizkfliiiiff :fit

* - free

mail f¥gyffxa¥?¥fhftgfhpgµbannfgn.

g. I = H ) Httl - 1) (1) + A) 6) + A) ( l ) = f to.

New basis: Eu, ,%} µ =

, ,

J' ' HK 3

↳ = Projuplg) = a - Proju, # = I - Higden, Estaing;D=kt÷tItl¥H

The Gram - Schmidt orthogonal nation Process

Given a basis Be = { vi. b. . .

, Vp } of a subspace V,

we can

fmd a new basis O = { 4,4 ,-

, up} with the properties :

4) O is orthogonal(2) span { vi. b

,. . ,b } =

span { % ↳ nun } for Is ksp .

i&. spam { b) = spank }

, span { a g) = span{ byal,

. ..

y ,= g

ya = a - Prejulb) = b- thigh y ,

↳ = Prej µ,at B) = g - Preju, B) - Projects) =

g- ifeng.in#gttfg

; '

i xp- Hindu - fifths -

. - ..

- Huffman .

The Gram - Schmidt Process takes a basis D= { %% . ,b}for a subspace V and turns it into an orthogonal bass Ofuyu. } .

,.u,}

ny = g ; it,= tthludl

.

↳ = b - G.b) I , ; th = GAY , 11

↳ = g- ( g. b) ud - lg.lk )lh ; the 43/114311.

or You can then produce an ortho normal basis 8=1 K,

I,

...

,In }

by normalizing .

Fist premise by↳,. .

, up ,then hamatze@ end

.

Eg .

Find an erthonormal basis for CDA whereA =/.}lg}g§|yet , =/!z| t.yyy.NET#tt=9iuIeftfH basis→ a ± I

lb = ¥ - b. Ul

t.gg#.u=lktfH=lkt:niitfrkln.ugyi;sr.

↳ = I -

thingy,bjuugp.us darts154=18

ftp.FH -

'

ftp..fi#.iui=rsEt. " ↳µ%i. 114311=3 Nz

Question : How can you heaver the original vectors { %%% ... }

from the new on.

basis { In,

Is,

B,

... ] ?

it '

then then,

→ I =

indiumni = * ,in - ixianuit . "¥yk+

them±,

as = huts ,( b

- 19 # HI - Nikki ) →ganglia

,+ Koh

43 + I g. b) th

/ st.is/=/uikB/frgryrE.- R

M ⇒ Q

Theorem : If A is an mxw matrix with linearly independent

columns,

then A has a factorization

A=QR←upper - Star

QTQ=I → a R_ orthermal- 61 mnj

O - 3 I 2Es .

pz§f) = EB %.YK (our "4) O o 31k

What is this good for ?

Calculating ogenoatues.

A = QR ; A,

= RQ