Lecture 21: Quick review from previous lecture

• The Gram matrix is

– In Rn, Gram matrices are always positive semidefinite

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

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

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

MATH 4242-Week 9-2 1 Spring 2020

K > O, positive definite

① K -- KT

② XT Kx >0.torallx.to.

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

-

3.6 Complex Vector Spaces

To finish Chapter 3: let’s briefly discuss complex vector spaces and complex innerproducts.

Recall that a complex number as an expression of the form

z = x + iy, x, y 2 R.

The complex conjugate of z = x + iy is

z = x� iy.

Thus,|z|2 = zz

• Everything we have done in Chapter 1 with Rn and real-valued scalars worksin Cn with complex-valued scalars.

• In particular, Gaussian elimination works exactly the same way if the numbersare in C instead of R.

MATH 4242-Week 9-2 2 Spring 2020

O

-

i -- Fi.

i'-- f-if

-

= - l.

-

nm

=

=

2- = It I.

( It ft ).

E = I - i. ft

12-12 = Z E = Citi I

= I - i'= It 1=2

.

✓ 171=1 Itil = F.

- #

In general , z = x tiy .

12-1-

= x't y'

,lzl = Tty .

§ Inner product on Cn:

hw, zi =nX

i=1

wizi

• This way, hz, zi =Pn

i=1 zizi =Pn

i=1 |zi|2, which is positive if z 6= 0.

• Note that hw, zi is not symmetric; rather it is conjugate-symmetric:

hw, zi = hz,wi

• For c, d 2 C,hcu + dv,wi = chu,wi + dhv,wihu, cv + dwi = chu,vi + dhu,wi

MATH 4242-Week 9-2 3 Spring 2020

A

÷??" 1h15 = ( V.V) 20 ; CV

,V> =O ⇒ V=O

.

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

Find Hull.

Hull =€i¥t3F 1x+iyf=x'ty

=#lzi F- 04222 t 4 t 9 = 4

.

= 15T' #

4 Orthogonality

4.1 Orthogonal and Orthonormal Bases

We’ve already seen that in an inner product space V , two vectors x and y are saidto be orthogonal if hx,yi = 0.

Fact: If v1, . . . ,vn are nonzero vectors that are “mutually orthogonal”, mean-ing hvi,vji = 0 if i 6= j, then v1, . . . ,vn are linearly independent.

Definition: Suppose v1, . . . ,vn are nonzero vectors that are mutually orthog-onal. If additionally kvik = 1, we say v1, . . . ,vn are orthonormal.

Definition:

• If v1, . . . ,vn are mutually orthogonal vectors that are also a basis for V (sodimV = n), we say they are an orthogonal basis.

• If v1, . . . ,vn are orthonormal and a basis for V , we say they are an or-

thonormal basis.

Example. In Rn equipped with the standard dot product, an orthonormal basisis the standard basis:

e1 =

0

BBBBBBB@

100...00

1

CCCCCCCA

, e2 =

0

BBBBBBB@

010...00

1

CCCCCCCA

, en =

0

BBBBBBB@

000...01

1

CCCCCCCA

MATH 4242-Week 9-2 4 Spring 2020

murmur

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

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

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

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

C , CV, ,V , 7=0

Call 4112=0 =3 C, = O . Similarly , we can getCz =O, --- ,Cu=O

① ②- -

① ②

( thrill = - -. = Krull = l )

.

= I

- -

- - "-

① Lei,er ) = O , -- - , Cei , Ej > = O , Isis

's h.

② It ejll = I , is j E n .

Fact: Suppose that v1, . . . ,vn are nonzero, mutually orthogonal (resp. or-thonormal) vectors. Then v1, . . . ,vn form an orthogonal (resp. orthonormal)basis for their span W = span{v1, . . . ,vn}.

Fact: Suppose that v1, . . . ,vn is orthogonal basis. Then

v1

kv1k, . . . ,

vn

kvnkform an orthonormal basis.

Example. Explain why vectors x = (1 1 0)T , y = (1 �1 1)T , and z =(1 �1 � 2)T form an orthogonal basis in R3? Turn them into an orthonormalbasis.

MATH 4242-Week 9-2 5 Spring 2020

t.ru,

E " "" " I orthogonal (orthonormal,

basis for W .

*

-

- IHIT,,H= "II 11911=1 .

X. y , 't are mutually orthogonal . lx.y.tl linearly<x. y > = ( ( b )

,( ÷ ) ) = O

.

independence

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

.

(y , Z ) = O.

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

.

life . En,¥, 1=1 # l !) , rift ) .#ID

.

is an orthonormal basis for IR?

§ 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)

basis. However, in general this is not so easy.

Fact: If v1, . . . ,vn is an orthogonal basis in any inner product space V , thenfor any vector v 2 V we have

v = a1v1 + · · · + anvn,

where

ai =hv,viikvik2

, i = 1, · · · , n.

Moreover, we have

kvk2 = a21kv1k2 + · · · a2nkvnk2 =nX

i=1

✓hv,viikvik

◆2

.

[To see this:]

MATH 4242-Week 9-2 6 Spring 2020

-mein

-

-

( t )

ft-

-

(2)

'1) For any r EV , we can write

✓ = A , V, t aah t.- -t Anon .

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

,r, ?

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

, ar,v, > = a , 114112

.

⇒ a , = Cv, v, >.

Sanitary , we can get as = %- . - -- ,An -_ LV,

Vn) #. # * Cai

, as , . . , an ) is coordinate

(2) Exercise.

I of vector V in basis Eri,. . .ru ?

Example. Consider the orthogonal basis x = (1 1 0)T , y = (1 �1 1)T , andz = (1 �1 � 2)T of R3. Write v = (1 2 3)T as the linear combination of x, yand z.

Example.

(1) The basis 1, x, x2 do NOT form an orthogonal basis.

(2)

p1(x) = 1, p2(x) = x� 1

2, p3(x) = x2 � x +

1

6.

is an orthogonal basis of P (2).

(3) Write p(x) = x2 + x + 1 in terms of the basis p1, p2, p3 in (2).

MATH 4242-Week 9-2 7 Spring 2020

f- 4Yi x + 9hyt4i¥z .

< r,x > = ( (

'

g ),f ! ) > = I -12=3

11×112 = C ( lol,( j ) ) = 2

.

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

.

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

.

( V, Z )= - 7

,

112112 = 6.

Thus,

E Ext Ey - Iz#

for R'"

( co , D )

( I,X > = fo' Ix dx = EM! = % to

,

<x. I > = go'x y' dx = f'd dx = #to-Icant#day.

( Con) ).

Text.

4190 )-