MATH 2022 Linear and Abstract Algebra

LECTURE 33 Thursday16/05/2019-

Developing more techniques / results for inner product spaces-

- notions of length and angles between vectors

a-arisenaturallybyrearraugingthecauchy-schwartinequa.li

- notions of orthogonality or perpendicularity9

criticaltosolvingmiuimisatiowprobleu- utility of orthonormal bases for vector spaces

a

facilitatingfindingcoordinatesofre.at

Developing more techniques / results for inner product spaces-

- notions of length and angles between vectors

a-arisenaturallybyre-rraugiugthecauchy-schwartinequa.li

- notions of orthogonality or perpendicularity9

criticaltosolvingmiuimisatiowprobleu- utility of orthonormal bases for vector spaces

a

facilitatingfindingcoordinatesofve.at

- able to recover geometric properties such as the

friaagleine#g and a generalised Theorem of Pythagoras-

Recall that a real vector space V is an innerproduet-pae.it

:ii÷:::÷::g⑦ HI, I EV I C t tf R) CHE,

IS = ICE,

I )

Recall that a real vector space V is an innerproduet-pae.it

:ii÷÷:::÷i::y⑦ HI, I EV I C the R) CHE,

IS = ICE,

I )

and it follows that, for all y , I

, I f V,

X f IR,

< us , Itf ) = LI , I ) t LI , I ),

< I , XI ) = X Cf , I ),

LE,

on > = Con,

I ) = O.

Recall,

we define the length of I EV to be

11 Ill = TEES

and then we get ( after considerable effort )

-cauchy-schwarzinegnality.it to, I 's I E HellHell

.-Hence

,I LE

,I > I

- E I11 Ill Il Ill

So thatCE

,I )

- i E - I I11 Ill Hell

Recall,

we define the length of I EV to be

11 Ill =

and then we get ( after considerable effort )

-cauchy-schwarzinegnality-ilco.ws 's I E HellHEH.-

Hence, lce.IS/- s ,

so that - i s I I11 Ill 11111

11 Ill 11111 I

and there is a unique Of Co,IT ] such that

cos O =( I ,

I )

¥11111

and we call O the angie between I and I .

Hence, lce.es/- s ,

so that - i s I I11 Ill 11111

11 Ill 11111 I

and there is a unique Of Co,IT ] such that

cos O =( I ,

I )

Fell

and we call O the angie between I and I .

Rearranging this,

we get

-( I,

I > = HEH HIM cos O

-

coming full circle to recover a" geometric

" definition.

Exampled : Let the inner product on1124 be the usual

dot product ,

and put

it = C - I,

- I,

I,

I ),

I = ( 3,5 ,I

,I )

,I = ( 6

,-4

, 1,1 ).

Then( y , I ) = - 3 - T ti ti = - 6

,

Hunk = tt = 54 =L,

11511 = Its = 536 = 6,

and indeed ✓

I LI, I > I = I -61=6 E 12 = Hell HEH

.

We haveCe

,I ) - 6

- = -= -

I

Hell KILL 2×6 2I

so the angle between y and I is.

However, < us ,

I ) = - 6+41-1+1 = 0,

and( I

,I ) = 18 - 20 t It ) = O

,

so the angles between y and if and between I and if

are both The,

So

I I I and It @ .

In

general,itcI,I7=othenO=Tzanwe say that I and I are orthogonal ormutuallyperpen-d.caA set of vectors in an inner product space is

called orthogonal if every pair of distinct vectors

is orthogonal .

It,

further,

the length of each vector is I,

then

the set is called orthonormal-

Call a vector I normal or normalised if 11 Ill =/.

If Efg thew KIK to and we may normalise r

forming I- u by

I -- Era I -

- ÷ ,-

so that HE 11=1.

In

general,itcE,I7=othenO=Tzanwe say that I and I are orthogonal or-mutuallyperpen-d.ca

A set of vectors in an inner product space is

called orthogonal if every pair of distinct vectors

is orthogonal .

If,

further,

the length of each vector is I,

then

the set is called orthonormal-

n

Call a vector I normal or normalised if 11 Ill =/.

Some examples of orthonormal sets-

:

Ci) { in, I } = { 4,07

,Coil ) } E R

"

Cii ) { I, I ,

k } = { a. aol.com , ol,

Co, on ) } E IR

'

Ciii) { C1,010,07,

Coil , 0,01,

( 0,0 , 1,0 ),

( 0,0 ,o

,, ) } E IR 't

Civ ) { Coiled,

C - E,

o,

# I,

( rt,

o,

rt ) ) ER'

a { ¥ , IT , 9¥,

" '

,. - -

,

"E . ÷ .

. - - } Ev

where ✓ is the space of continuous functions : fit,AT -7 R

with inner product s f, g) = fit fcnlgcn ) die

.

An orthonormal basis because it reduces-

is useful

the problem of finding coordinates to calculations

of inner products :

proposikou-rletB-sb.br#a

orthonormal basis for V.

Then,

for I C- V,I=(E,k,7k,tCE,ke)ket---tCI,kn)b

Proof ..

Let Itv,

so

I = X, I ,

t X, brat - - - t Xu bun

for some X, ,

Xu,

. . .

,Xn f IR

.

Then,

for i =L,

. . .

,n

,

< I ,ki ) = L X

,k

,t - - - t Xnkn

,Ii )

= t, Cbn

, Ei ) t. - - t hit C ki

. , ,ki ) t Xi LEI

,ki )

+ Xin L kin,

ki) t - - - t Xu tbn , Ki >

= X,

lol t . . - t ki. ,

Co) t hill ) t hit ,Co) t - - - t tuco )

becansebni-sknanareqaifjyorthog.ua#

= o t . . .to t hi to t - - to = Xi

,

and the result follows. D

Exampte : Let B = { I , ,In

,Is } where

I ,= ( oil ,

o ),

In = ( - 45,

o,

%-),

I ,= ( 315,0 ,

4/5 ).

Thew B is an orthonormal basis for IRS.

LetI = L 7

,- I

,8)

.

Thew( I

,E , ) = o - I to = - I

,

CE , en ) =- ¥ to + ¥ = - Is

,

( I,

Is ) = ¥ to t = 515 .

Exampte : Let B = { I , ,In

,Is } where

I ,= ( oil ,

o ),

In = ( - 45,

o,

%-),

I ,= ( 315,0 ,

415).

Thew B is an orthonormal basis for IRS.

↳ tI = ( 7

,- I

,8)

.

much less

" -

::÷:::¥ :*.

÷÷÷÷( I

,Is ) = ¥ to t = 5¥ .

equations

Hence \I I = - I ,- ¥ Eat Is

-

so CAB = %) .

"""""°t"÷%"÷"Ci) HEH do

Cii ) Il XIII = 14111111 ( ther )

ciiisnetmswII.mg?!eqn.iit#Proofs

: Ci ) HEH = 5 30

and 11111=0 ⇒ T÷ = o ⇒ C I ,I 7=0

⇒ I = g. ✓

Cii )if YER Thew

11 tell =

= Ty'

CE, I >

= RE Ee

= 1×1 HEH. ✓

Ciii ) Observe that

11 It Ill'

= LITE, Ete )

= LE,

I ) t CE , I ) t CI,

I ) t CE , I )

= Hell'

t CE, I ) t CE , I ) t HILT

= Hell'

t 2 LI,

I ) t HEH"

E Hell 't 21 LE,

I > I t Hell'

E Il Ill 't 21111111111 t HIM

= ( Hell + ay ,,yLbycauehy-su.ua#

Ciii ) Observe that

11 It Ill'

= LITE, Itv )

= LE,

I ) t CE , I ) t CI,

I ) t CE , I )

= Hell 't CE, I ) t CE , I ) t Hell

-

= HEH 't 2 LI,

I ) t HEH"

E Hell 't 21 LE,

I > It Hell'

I 1111ft 21111111111 t 111112

= ( Hell + ay ,,yLbycauehy-su.ua#

i. e . Hft I 11'

f ( HEH t 11111 ) ?"

HE tell E HIM t HIM.

✓ M-

We also have the

^""""&T↳""°t%th%°XSuppose C E,

17 = o.

ThewHEtIlT=KElTtHE€

n¥÷ItI

Proof :

my+ I y'

= CITE,

It I ?

= HEH 't 2 CE,

I ) t Hell'

asinpreviouspro= Hell 't do ) t 111112

i"eEu= HEH 't HIM

✓ I

Exampled : I = ( 3,5 ,I

,I )

,I = C 6

,-4

,I

,I ) C- Rt

.

Then< I ,

I ) = 18 - w ti ti = o,

It I = ( 3,5 , 1,1 ) t 16,

-4,

I,

I ) = ( 9,

I,

2,

2) .

Observe that

flirty It'

= 97ft it 22=81 Htyty = 90,

HEH'

= 37575+5=9+471 ti = 36,

Hugh"

= 67C -477171'

a 36+16+11-1=54,

and indeed HEH 't Hell ! 36+54--90=111+111 ?✓