mat423 lecture topic 4 vector.pptx

8/17/2019 MAT423 LECTURE TOPIC 4 VECTOR.pptx

1/79

4.1 Introduction toVectors

Real vector spaces &Subspaces

Linear Algebra I

TOPIC 4 :

VECTOR SPACES

Assoc. Prof. Dr. Khalipah Ibrahim


2/79

R1 = R = { a1, b1, … }

R2 = { (a1 , a2), (b2 , b2), … }

R3 = { (a1, a2, a3), (b1, b2, b3), … }


Defn: If n is a positive integer, then an ordered n-tp!e is

a se"en#e of n rea! n$bers (a1 , a2 ,…,an)% &he set ofa!! n-tp!es is #a!!ed n-spa#e and is denoted b' Rn%

Ordered n-tuple

2


3/79

Defn: et ve#tors u = (1,2,…,n) and v = (v1,v2,…,vn) in

Rn%

1) u=v if 1=v1, 2=v2,…, n=vn%

2) $ of u and v is u*v =(1*v1, … , n*vn)

3) #a!ar +!tip!e of a s#a!ar and u is u= (1,2,…,n)

) .u = (-1 , -2 ,…,-n) is the negative of u%

Standard Operations on Rn

Assoc. Prof. Dr. Khalipah Ibrahim3

/ero ve#tor, 0 = (0, 0, …, 0)


4/79

&h$ %1%1

If u = (1,2,…,n) , v = (v1,v2,…,vn) and = (1,2,…,n)are ve#tors in Rn and and $ are s#a!ars, then:a) u * v = v * u b) u *(v * ) =(u * v) * #) u*0 = 0*u = u d) u * (-u) = u - u = 0

e) ($u) = ($)u f) (u * v) = u * vg)(*$)u = u * $u h)1u = u


Properties of Vector Operations in Rn

4


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&h$ %1%2If u, v and are ve#tors in Rn and is an' s#a!ar, then:a) u v = v u b) (u * v) = u * v

#) (u) v = (u v)d) v v 0%

v v = 0 i4 v = 0


Euclidean Inner Product

!


Rn%

5#!idean inner prod#t u v is de6ned b'

u v =1v1*2v2* … *nvn


6/79


Norm & Distance in Euclidean n-Space

"


Rn%

1) 5#!idean 7or$ or ength of a ve#tor :

2) Distan#e bet to ve#tors :

( 2n22

21

21 +++=•= ...u)uu #

vvv

))-()-((d2

222

222

11

182

)$...)$)$

vuvuvuv)$u%

−++−+−=

•=−=


7/79

Defn: et 9 be an arbitrar' none$pt' set of obe#ts ith

the de6nes addition ; s#a!ar $!tip!i#ation%

et u,v, ∈9 and ,$ are s#a!ars% 9 is a ve#tor spa#e if

a!! u,v,,,$ satisf' a!! of the fo!!oing 10 a) u ∈ 9

?) (u * v )=u * v @) (*$)u=u*$u

A)($u) = ($)u 10)1u=u

Vector Space Axioms



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&h$ %1%1 et 9 be a ve#tor spa#e, u ∈9 and a s#a!ar,

then,

1) 0u=0

2) 0=0

3) (-1)u = -u

) If u = 0, then = 0 or u = 0

Vector Spaces

'

5


9/79

Defn: bspa#e of a ve#tor%

et 9 be a ve#tor spa#e and is a sbset of 9%

is #a!!ed a sbspa#e of 9 if is a ve#tor spa#e nder

the addition and s#a!ar $!tip!i#ation de6ned on 9%

Subspaces

(

&h$ %2%1% If is a set of one or $ore ve#tors fro$ ave#tor spa#e 9, then is a sbspa#e of 9 if and on!' if

the fo!!oing #onditions ho!d%

1) If u and v are ve#tors in , then u * v is in

2) If is an' s#a!ar and u is an' ve#tor in , then u is in

%


2 Eondition for a set of ve#tors, to be a sbspa#e of a

ve#tor spa#e 9%

1) For ever' u%v∈, u*v ∈


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*+ ,A- 13 4aii

Let

/ec condition 1


on2$on$

! G

C-'2v


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*+ ,A- 13 4aii

/ec condition 2


f!6!!e2%#onditionB &hs

(C)-(')2(v) C'v2

C'v2<

C'<

v

C'<

vB

∈

+= −+=

−+=

=

=

)$


12/79

*+ ,A- 13 4b

LS

RS

$A5)6A5

7us% te a8io9 :ails to old


++=+

dh#g

f beaGB )$

+

=+ hgf e

d#

ba

GB

++=dh#g

f bea

+

= hg

f e

d#

ba

++=dh#g

f bea

++

= dh#g

f bea22

++

≠ dh#g

f bea


13/79

*+ ,A- 12 4a


Rv,,and0v,70

v G

Rr",p,and0"p,70r

"pB

∈≥+∈

=

∈≥+∈

=

++

++=

00r

v"p

+

=+

0

v

0r

"pGB

)$)$)$)$ v"pv"p +++=+++

Let

/ec /ondition 1

/ondition 1 is :ul;lled

/ec condition 2

/ondition 2 is not satis;ed. - is not a subspaceo:


14/79

*+ ,A- 12 4b

Let p$8)=ab8c82

LS

$9)p$8) = $9)$ab8c82)

= a$9)b8$9)2c82

RS

p$8)9p$8)=$ab8c82)9$ab8c82)

= $ab8 2c82) $a9b892c82)

= 2a$9)b8 2 c8292c82

6 $9)p$8)

6LS

7e a8io9 is not satis;ed



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*+ Apr 2011a


Let u = $a%b%>) and v = $d%e%:)

LS c$uv) = c$ad%be%>:)

=$ad%c$be)%?$>:))

=$ad% cbce%?>?:)

RS cucv =c$a%b%>)c$d%e%:)=$a%cb%?>)$d%ce%?:)

=$ad% cbce%?>?:)

c$uv)=cucv. 7us% te given a8io9 olds.


16/79

*+ Apr 11 4b


Let p = r s8 t82 ∈ ere rt =?2

@ = uv8 82∈ ere u=?2

/ec condition 1p@ = $r s8 t82) $uv8 82)

=$ru) $sv)8 $t)82

$ru) $t)=$rt) $u)

= ?2 $?2)=?46?2p@∉

7us% condition 1 does not old

Set is not a subspace o: 2


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*+ B/7 10 3a


C414%001

441 A)i ∉

=

=

∈

=

=

=

000

000


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*+ B/7 10 3a


/ec condition 1

$ap)$cr)=$ac) $pr)

= b @

7us% b@=$ap) $cr)

A5∈

C/ondition 1 is satis;ed

b=$ac) =ac

A∈C. 7us% C is a suspace o: <23

Crp@%00s

r@p5

Ccab%

00d

cbaLet Aiii)

∈

=

∈

=

00sd

rc@bpa

00s

r@p

00d

cba 5A

=

=

00d

cba

00d

cba A

=

=


19/79Assoc. Prof. Dr. Khalipah Ibrahim


20/79

4.2 Linear/o9bination

Linear Algebra I

TOPIC 4 :

VECTOR SPACES



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5g 1% Hiven ve#tors u = (1,2,-1) and v = (>,,2)

deter$ine hether ea#h of the fo!!oing ve#tors is a

!inear #o$bination of u and va) = (A,2,?)

b) 8 = (, -1,@)

Defn: Eonsider a ve#tor spa#e 9% If ∈ 9, then is #a!!ed

a linear co9bination of the ve#tors

v1, v2, … , vn in 9 if it #an be ritten as

= 1v1*2v2* …*nvn

here 1, 2, … , n are s#a!ars% &hese s#a!ars are #a!!ed

the #oe#ient of the !inear #o$bination%

Linear ombination

21Assoc. Prof. Dr. Khalipah Ibrahim


22/79

*8a9ple 1 solution


2=2, 1=-3

= -3 * 2

is a !inear #o$bination of and v

&he s'ste$ is not #onsistent% 7o va!e of r1 and r2 to

rite

< = r1*r2v

< is not a !inear #o$bination of and v

−−

− 0

2

9

00

10

61

16

16

9

80

80

61

7

2

9

21

42

61

)a

−−

−

− 7

2

9

00

10

61

16

9

4

80

80

61

8

1

4

21

42

61

)b


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*8a9ple 1 solution



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&h$ %2%3 : If = {v1, v2, … , vn} is a none$pt' set of

ve#tors in a ve#tor spa#e 9, then:a) &he set of a!! possib!e !inear #o$binations of

ve#tors in is a sbspa#e of 9%

Linear /o9bination



25/79

5


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*8a9ple 2 solution

2"


27/79

*8a9ple 2 solution

2


28/79


29/79

4.3 Linearindependence

Linear Algebra II

TOPIC 4 :

VECTOR SPACES



30/79

&h$ %3%1 et be a set ith to or $ore ve#tors%

a) et is !inear!' dependent i4 at !east one of the

ve#tors in is e


31/79

&h$ %3%2

a) B ;nite set o: vectors that #ontains the Cero ve#tor

is linearlH dependent.

b) B set ith e


32/79

5,-1) and=(3,2,1) for$ a !inear!' dependent or !inear!'

independent set%

o!tion:

et #1 * #2v * #3 = 0For$ a !inear s'ste$ and so!ve

Linear Independence



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4.4 5asis andi9ension o: a

Vector SpaceLinear Algebra I

TOPIC 4 :

VECTOR SPACES



34/79

5


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3!

o!tion:

*8a9ple ! Solution


36/79

tandard basis for Rn%

R2 : ={(1,0) , (0,1)}

R3: = {(1,0,0), (0,1,0), (0,0,1)}

R: = {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}

--------------------------------------------------

Rn

: ={(1,0,…,0), (0,1,…,0),…,(0,0,…1)}

Standard basis :or n?space Rn

3"


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Standard basis :or olHno9ials% n

tandard basis for Kn

K1: = {1,


38/79

Standard basis :or


39/79

Defn: B nonCero ve#tor spa#e 9 is #a!!ed ;nite?di9ension if it #ontains a 6nite set of ve#tors {v1, v2, … ,

vr} that for$s a basis%

If no s#h set e


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1) di$ ( Rn ) = n

2) di$ ( Kn ) = n*1

3) di$( +$


41/79

&h$ %%

et a none$pt' set of ve#tors in a ve#tor spa#e 9%

1) If is a linearlH independent set, and v is a ve#tor

in 9 that is otside of span(),

then the set ∪ {v} that res!ts b' inserting v into

is sti!! linearlH independent%2) If v is a ve#tor in that is e


42/79

&h$ %%>

et be a 6nite set of ve#tors in a 6nite-di$ensiona!ve#tor spa#e 9%a) If spans 9 but is not a basis for 9, then can be

reduced to a basis for 9 b' re9oving appropriateve#tors fro$ %

b) If is a linearlH independent set that is not a!read'a basis for 9, then #an be enlarged to a basis for 9b' inserting appropriate ve#tors in to %

&h$ %%If 9 is an n-di$ensiona! ve#tor spa#e, and if is a set in 9ith e


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5%

a) Eonsider a set of ve#tors

= {(1,0,0,0), (-1,1,0,-1), (3,2,0,-2), (0,-1,-2,2),

(1,0,1,0)}

i% ho that the set spans R

ii% Deter$ine ve#tor(s) in that #an be re$oved sothat #an be a basis for R%

b) Eonsider the sets {


44/79

*8a9ple " Solution

44

* l " S l ti


45/79

*8a9ple " Solution

4!

* l " S l ti


46/79

*8a9ple " Solution

4"

/oordinate Vector o: a vector relative to a


47/79

5


48/79

4'

*8a9ple Solution

TOPIC 4


49/79

4.! Inner productSpaces

Linear Algebra II

TOPIC 4 :

VECTOR SPACES


Inner roducts


50/79

Defn: Bn inner prod#t on a rea! ve#tor spa#e 9 is a fn#tionthat asso#iates a rea! n$ber 〈u%v〉 ith ea#h pair of

ve#tors u and v in 9 s#h that it satis6es the fo!!oinga


51/79

5


52/79

*8a9ple 1solution

!2

Lengt and istance in Inner roductS


53/79

Defn:If 9 is an inner prod#t spa#e, then the nor$ (or !ength )

of

a ve#tor in 9 is de6ned b'

B unit vector is a ve#tor that has nor$ 1

&he distan#e beteen to points (ve#tors) and v isde6ned b'

d(u%v) =

Spaces

!3

2#1u%uu =

vu

Lengt and istance in Inner roduct


54/79

5


55/79

*8a9ple 2solution

!!

*8a9ple 2solution


56/79

*8a9ple 2solution

!"

*8a9ple 2solution


57/79

*8a9ple 2solution

!

Inner roduct o: :unctions


58/79

Inner Krod#t on ENa,bO.et : = f(


59/79

4." Brtogonal 5asesand Jra9?Sc9idt

rocess

Linear Algebra II

TOPIC 4 :

VECTOR SPACES


Brtogonal and Brtonor9al 5ases :or


60/79

De6nitions:1) &o ve#tors u and v in an inner prod#t spa#e are

#a!!ed orthogona! if 〈u,v〉 = 0%

2) B set of ve#tors in an inner prod#t spa#e is #a!!ed anorthogona! set if a!! pairs of distin#t ve#tors in the setare orthogona!% Bn orthogona! set in hi#h ea#h ve#tor

has nor$ 1 is #a!!ed orthonor$a! set%3) B basis of an inner prod#t spa#e that #onsists of

orthonor$a! ve#tors is #a!!ed an orthonor$a! basis% Bnorthogona! basis is a basis that #onsists of orthogona!

ve#tors%

Inner roduct Space

"0

&h$ >%3%3

If = {v1, v2, … , vn} is an orthogona! set of nonCero

ve#tors in an inner prod#t spa#e, then is !inear!'

independent%

/oordinates relative to Brtonor9al 5ases


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&h$ >%3%1 If = {v1, v2, … , vr} is an orthonor$a! basis

for an inner prod#t spa#e 9, and u is an' ve#tor in 9,

then,u = 〈u,v1〉 v1* 〈u,v2〉v2 *…* 〈u,vn〉vn%

〈u,v1〉, 〈u,v2〉, …, 〈u,vn〉 are the coordinates of u re!ative

to %(u) = (〈u,v1〉, 〈u,v2〉, …, 〈u,vn〉) is the coordinate vector

of u re!ative to %

&h$ >%3%2% If is an orthonor$a! basis for an n-

di$ensiona! inner prod#t spa#e, and if () = (1, 2, … ,

n) (v)= (v1, v2, … , vn), then

/oordinates relative to Brtonor9al 5ases

"1

2n2221 (a) +++= ...u2

nn2

222

11)v()v()v(),(d(b) −++−+−= ...vu

nn2211vvv,(#) +++= ...vu

/oordinates relative to Brtogonal 5ases


62/79

5


63/79

Solution

*8a9ple 3solution

"3

*8a9ple 3solution


64/79

*8a9ple 3solution

"4

Brtogonal pro>ection


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&h$ >%3% Kroe#tion &heore$

If is a 6nite-di$ensiona! sbspa#e of an inner prod#t

spa#e 9, then ever' ve#tor u in 9 #an be e

"!

Ort!o"onal Pro#ection


66/79

&h$ >%3%>5ver' nonCero 6nite-di$ensiona! inner prod#t spa#e hasan orthonor$a! basis%

&h$ >%3%

et be a 6nite-di$ensiona! sbspa#e of an inner prod#t

spa#e 9%a)If {v1, v2, … , vn} is an orthonor$a! basis for , and u

is an' ve#tor in 9, then

Kro u= 〈u,v1〉v1* 〈u,v2〉v2 *…* 〈u,vn〉vn%

b) If {v1, v2, … , vr} is an orthogona! basis for , and u is

an' ve#tor in 9, then

""

r2r

r22

2

212

1

1 v

v

v%u...v

v

v%uv

v

v%uupro>

Brtogonal ro>ections


67/79

5


68/79

"'

Jra9?Sc9idt rocess


69/79

Jra9?Sc9idt rocess is used to convert anH basisto an ortogonal basis.et 9 be an' nonCero 6nite-di$ensiona! inner prod#tspa#e, and {u1, u2, …, un} is an' basis for 9%

Bn orthogona! basis {v1, v2, …, vn} for 9 #an be fond as

fo!!os:

tep 1: et v1 = u1tep 2:Eonstr#t a ve#tor v2 that is orthogona! to v1:

tep 3:Eonstr#t a ve#tor v3 that is orthogona! to v1 and

v2 :

"(

12

1

1222

vv

v%uuv −=

222

2312

1

1333 v

v

v%uv

v

v%uuv −−=

Jra9?Sc9idt rocess


70/79

Solution.

Eontine in the sa$e $anner, after n steps an orthogona!set of n ve#tors {v1, v2, …% , vn} i!! be obtained%

5


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p

1

*8a9ple !solution


72/79

p

2

TOPIC 4 :


73/79

4. /ange o: 5asis

Linear Algebra II

TOPIC 4 :

VECTOR SPACES


/ange o: 5asis


74/79

et G = {u1, u2, … , unK and G = {u1

, u2P, … , un

} are

the bases for a ve#tor spa#e 9 and v ∈ V%If :

1)NvOG is the #oordinate ve#tor of v re!ative to G

2)NvOG is the #oordinate ve#tor of v re!ative to G

3)K is a transition $atri< fro$ G to G

then

NvOG = K NvOG

&ransition $atri< K #an be e


75/79

5Eonsider the bases 5 ={u1,u2} and 5 ={u1,u2} for R

2%

Hiven u1=(1,0), u2=(0,1), P1=(1,1) and P2=(2,1), 6nd:

a) &he transition $atri< fro$ G to GP%

b) &he transition $atri< fro$ GP to G%

#) NvOGP if NvOG =

!

&h$ >%%1

If K is the transition $atri< fro$ a basis 5 to 5P for a

6nite-di$ensiona! ve#tor spa#e 9, then K is invertib!e, and = K-1 is the transition $atri< fro$ GP to G%

=K-1 =NNuP1OG Q NP2OG Q … Q NPnOG O


−

3


76/79

*8a9ple "solution


77/79

p

*8a9ple "solution


78/79

'


79/79

*nd o: 7e8ts

7BI/ 4V*/7BR SA/*S

mat423 lecture topic 4 vector.pptx

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