Quick Review from Previous Lecture • the Gram Matrix Is

Home , Gram matrix

Lecture 21: Quick review from previous lecture

> O definite K , positive The Gram matrix is

- • ① K - KT

② XT Kx > 0.torallx.to.

( = > O Vi , > Vic C Vj , . nm Vj

– In Rn,Grammatricesarealwayspositivesemideﬁnite

– they are positive deﬁnite precisely when the vectors v1,...,vn are linearly independent.

————————————————————————————————— Today we will discuss ”Orthogonal(Orthonormal) bases”

————————————————————————————————— Lecture video can be found in Canvas ”Media Gallery” -

MATH 4242-Week 9-2 1 Spring 2020 3.6O Complex Vector Spaces To ﬁnish Chapter 3: let’s brieﬂy discuss complex vector spaces and complex inner products. Recall that a complex number as an expression of the form - '

- - i - i - Fi. f- z = x + iy, x, y . if - R 2 = - l . The complex conjugate of z = x + iy is - z = x iy. Thus, z 2 = zz nm| | Everything we have done in Chapter 1 with Rn and real-valued scalars works • = in Cn with complex-valued scalars. = In particular, Gaussian elimination works exactly the same way if the numbers • are in C instead of R.

It ft 2- = It I ( ) . .

= I - i E . ft I 12-12 = Z E = Citi ' = = - i It 1=2 I .

171=1 Itil = F ✓ . - #

= x ti z . In general , y - ' = x lzl = 12-1 't . y , Tty

MATH 4242-Week 9-2 2 Spring 2020 Inner product on Cn: § n w, z = w z h i iAi i=1 X This way, z, z = n z z = n z 2,whichispositiveifz =0. • h i i=1 i i i=1 | i| 6 Note that w, z isP not symmetric;P rather it is conjugate-symmetric: • h i w, z = z, w h i h i For c, d C, • 2 cu + dv, w = c u, w + d v, w h i h i h i ÷u,cv +?dw = c u, v +?d u, w h i h i h i

V=O " =O ⇒ 20 CV V . 1h15 = ( V.V ) ; , >

' zi Find Hull . : Iti -35 . EI v= ( , ,

Hull = €i¥t3F 1x+iyf=x'ty lzi F- 0422 = 2 t 4 t #9 = 4 .

= 15T ' #

MATH 4242-Week 9-2 3 Spring 2020 4 Orthogonality 4.1 Orthogonal and Orthonormal Bases We’ve already seen that in an inner product space V ,twovectorsx and y are said to be orthogonal if x, y =0. h i Fact: If v1,...,vn are nonzero vectors that are “mutually orthogonal”, mean- ing vi, vj =0ifi = j,thenv1,...,vn are linearly independent. h i 6 murmur

: = this V . . To see Consider c t . O [ ) , , + Carn .

- - - = To show C =O , Cn O We want , , . -

. . - = , t t ( O V, > ( CY Cnvrtv , > , .

= . . - O Be . . tcf Gcr ,u ,> tacru.is

CV C, , ,V , 7=0 we can Call =3 C = O , get 4112=0 , . Similarly =O Cz , --- ,Cu=O Deﬁnition: Suppose v1,...,vn are nonzero vectors that are mutually orthogonal. If additionally v =1,wesayv ,...,v are orthonormal. k ik 1 n Deﬁnition: ① ② If v1,...,vn are mutually orthogonal vectors that are also a basis for V (so • - - dim V = n), we say they are an orthogonal basis. ① ② If v ,...,v are orthonormal and a basis for V ,wesaytheyareanor- • 1 n = - - = ( thrill . = Krull l thonormal basis. ) .

Example. In Rn equipped with the standard dot product, an orthonormal basis = I is the standard basis: 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0

e1 = , e2 = , - - en = B . C B . C " B . C - - - B . C B . C B . C B C B C B C B 0 C B 0 C B 0 C B C B C B C B 0 C B 0 C B 1 C B C B C B C 's h O . @ A @ A - @ A = - = ① Lei er ) - Cei O Isis , , > , MATH 4242-Week 9-2 4 , , Ej Spring 2020

It = I is E ② ejll , j n . Fact: Suppose that v1,...,vn are nonzero, mutually orthogonal (resp. orthonormal) vectors. Then v1,...,vn form an orthogonal (resp. orthonormal) basis for their span W =span v ,...,v . { 1 n} , " " ( orthonormal E "" I orthogonal

t.ru, basis for W . *

Fact: Suppose that v1,...,vn is orthogonal basis. Then - v v 1 ,..., n v v k 1k k nk form an orthonormal basis. - I

11911=1 . HIT, ,H= "II

Example. Explain why vectors x =(110)T , y =(1 11)T ,andz = (1 1 2)T form an orthogonal basis in R3?Turnthemintoanorthonormal basis.

X. 't are . y , mutually orthogonal lx.y.tl linearly

< = = O independence x. y > ( ( b ) ( ÷ ) ) . ,

a. = 0 z > =L lol > . , ft ( . )

( Z = y , ) O .

in an basis IRS Ex Z is . Thus , y , ) orthogonal ,

life . , 1=1 # l ) ¥ ! , .# En , rift ) ID .

is an orthonormal basis for IR? MATH 4242-Week 9-2 5 Spring 2020 Computations in Orthogonal Bases § What are the advantages of orthogonal (orthonormal) bases? It is simple to ﬁnd the coordinates of a vector in the orthogonal (orthonormal) - me basis. However, in general this is not so easy. in -

Fact: If v1,...,vn is an orthogonal basis in any inner product space V ,then for any vector v V we have -2 ( t ) v = a1v1 + + anvn, ··· - where v, vi ai = h 2i,i=1, ,n. vi ··· -ft k k Moreover, we have n v, v 2 v 2 = a2 v 2 + a2 v 2 = h ii . (2) k k 1k 1k ··· nk nk v i=1 i X ✓ k k ◆ ' write 1) r EV we can [To see this:] For any ,

= t aah t . - - t Anon ✓ A , V, . #

- -- r = t a. t tank , v r ( air, v. , ? ( , , >

er. + = a. ,v , > adf.ir?7-.--ta.af?oD

Thus , = ar v a , 114112 , . , >

= Cv, v, > ⇒ a , .

= - -- we can as . , Sanitary , get %- Vn # LV, ) An -_ . # * . coordinate Cai as . an ) is , , ,

. MATH 4242-Week 9-2 6 V Spring. 2020 of vector in basis Eri, .ru (2) Exercise I ? . Example. Consider the orthogonal basis x =(11 0)T , y =(1 11)T ,and z =(1 1 2)T of R3.Writev =(12 3)T as the linear combination of x, y and z. f- x + 4Yi 9hyt4i¥z .

' = = I -12=3 < r x > ( ( > , g ) f ! ) ,

= = 2 11×112 C ( lol ( ) . , j )

' v = = -21-3=2 < > C I . , y ) ( g ) , ft )

= Nyit L = 3 l 't ) l 'T ) ) . ,

= - ( V, Z ) 7 ,

= 112112 Thus E - , 6 . Ext Ey Iz Example. #

2 " (1) The basis 1,x,x do NOT form an orthogonal basis.for R' ( co , D ) ' ( I X > = Ix dx = EM = % to ! , , fo

' ' = x dx = = go y f'd dx # to - (2) I cant 1 1 p (x)=1,p(x)=x ,p(x)=x2 x + . #day.1 2 2 3 6 (2) is an orthogonal basis of .( Con ) ) . P Tex t 2 (3) Write. p(x)=x + x +1intermsofthebasisp1,p2,p3 in (2).

- 4190 )

MATH 4242-Week 9-2 7 Spring 2020