Sec 4.2 + Sec 4.3 + Sec 4.4

Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces

Let V be a set of elements with vector addition and multiplication by scalar is a vector space if these operations satisfy the following:

Vector Space

numbers real are and vectorsare and if r, swu,v ve)(commutati ) uvvua

ve)(associati )()( ) wvuwvub element) (zero 00 ) uuuc

inverse) (additive 0)( ) uud

ive)(distribut )( ) rvruvure

)( ) suruusrf

)()r( ) urssug

)1( ) uuh

Example space vector a is nR

numbers real ,,,

where],,,,[

21

21

n

nn

vvv

vvvvR

Set:

],,,[

],,,[],,,[

2211

2121

nn

nn

uvuvuv

uuuvvvuv

Vector Addition:

Scalar Multiplication:

],,,[ 21 ncvcvcvcv

Example space vector a is nxnM

matricesnxn all ofset thenxnMSet:

nnnnnn

nn

nnn

n

nnn

n

uvuv

uvuv

uu

uu

vv

vv

uv

11

111111

1

111

1

111

Vector Addition:


nnn

n

cvcv

cvcv

cv

1

111



Vector Space





)( ) suruusrf

)()r( ) urssug

)1( ) uuh

Example space vector a is nP

ndegrewith

ploynomial all ofset thenPSet:

)()())(( xgxfxgf Vector Addition:


)())(( xcfxcf

Example space vector a is F

functions valuedreal ofset theFSet:

Vector Addition:


)()())(( xgxfxgf

)())(( xcfxcf

Linear combinationV space vector in the vectors threeare ,, 321 uuu

vectors three theofn combinatiolinear called is

following then the ucucuc 332211

Example

51

231u

v is a linear combination of u1,u2


64

1082u22in xM

221121

31ucucv

Example xxgxxf 22 cos)( ,sin)( Fin )2cos()( xxh f(x), g(x)h(x) ofn combinatiolinear a is

Example2324 2)( ,)( xxxgxxxf

4in P

342)( xxxh

f(x), g(x)h(x) ofn combinatiolinear a is

Linearly dependent vectorsV space vector in the ,,, 21 nuuu

are said to be linearly dependent provided that one of them is a linear combination of the remaining vectors


Example

51

231u

v is a linear combination of u1,u2

64

1082u22in xM

221121

31ucucv

Example xxgxxf 22 cos)( ,sin)( Fin )2cos()( xxh f(x), g(x)h(x) ofn combinatiolinear a is

Example2324 2)( ,)( xxxgxxxf

4in P

342)( xxxh

f(x), g(x)h(x) ofn combinatiolinear a is

{ u1, u2, v} are linearly dependent

{ f, g, h } are linearly dependent

{ f, g, h } are linearly dependent

otherwise, they are linearly independent

Linearly dependent vectorsSec 4.2 + Sec 4.3 + Sec 4.4

CHAPTER 4 Vector Spaces

???dependent or t independenlinearly

,

0

0

0

,

5

4

3

,

1

2

1

321

uuu

,

5

10

5

,

5

4

3

,

1

2

1

321

uuu

WronskianSec 4.2 + Sec 4.3 + Sec 4.4


abledifferenti-1)-(n functions-n be ,,, 21 nfffLet

)1()1(2

)1(1

''2

'1

21

nn

nn

n

n

fff

fff

fff

tdeterminannxn theisskian their wron

Example ,1)( ,)( ,)( 32

23

1 xfxxfxxfFin Find the wroskian

Example )2sin()( ,)( 22

1 xxfexf x Fin Find the wroskian

Example ,32)( ,3)( ,)( 332

31 xxxfxxfxxfFin Find the wroskian


functions dependent linearly -n

be ,,, 21 nfffLet

Example

,1)( ,)( ,)( 32

23

1 xfxxfxxf

)2sin()( ,)( 22

1 xxfexf x

,32)( ,3)( ,)( 332

31 xxxfxxfxxf

0WTHM:

???dependent or t independenlinearly

Wronskian



Subspace





)( ) suruusrf

)()r( ) urssug

)1( ) uuh

Example space vector a is 22xMV

0 such that

00

matrices 22 all ofset the

baba

x

W

Definition:

VWLet

VLet

ofsubset be

space vector a be VW

W is a subspace of V provided that W itself is a vector space with addition operation and scalar multiplication as defined in V


THM:

WvuWvu in then ,in , (1)


0 such that

00


baba

x

W

VW

W subspace of VTwo conditions are satisfied

WcuWu in then ,in (2)


01 matrices 22 all ofset the

baxW

Spanning setSec 4.2 + Sec 4.3 + Sec 4.4


} ,,, { n21 vvv span the vector space V if

every vector in V is a a linear combination of these k-vectors

Linearly Independent} ,,, { n21 vvv Linearly independent if the only solution for

02211 nnvcvcvc is ,0 21 nccc

} ,,, { n21 vvv Definition:

is a basis for the vector space V if

Vspan } ,,, { b)

tindependenlinearly } ,,, { a)

n21

n21

vvv

vvv

Example

0

0

1

1u

0

1

0

2u

1

0

0

3u

3321

321

Rspan } ,, { b)

tindependenlinearly } ,, { a)

uuu

uuu

3321 Rfor basis a form } ,, { Hence, uuu


} ,,, { n21 vvv Definition:

is a basis for the vector space V if

Vspan } ,,, { b)


n21

n21

vvv

vvv

Example

0

0

1

1u

0

1

0

2u

1

0

0

3u

3321

321

Rspan } ,, { b)

tindependenlinearly } ,, { a)

uuu

uuu

3321 Rfor basis a form } ,, { Hence, uuu

Example

00

011u

00

102u

01

003u

10

002u

Example ,)( ,)( ,1)( 2321 xxfxxfxf

2x24321

4321

Mspan } ,,, { b)

tindependenlinearly },,, { a)

uuuu

uuuu

2x24321 Mfor basis a form } ,,, { Hence, uuuu

2321

321

Pspan } ,, { b)

tindependenlinearly },, { a)

fff

fff

2321 Pfor basis a form } ,, { Hence, fff

Example ,1)( ,1)( ,1)( 23

221 xxxfxxfxxf

2321

321

Pspan } ,, { b)

2t independenlinearly },, { a)

fff

)-(W fff



Definition: The dimension of a vector space V is the number of vectors in any basis of V

Example

0

0

1

1u

0

1

0

2u

1

0

0

3u

3321 Rfor basis a form } ,, { uuu

Example

00

011u

00

102u

01

003u

10

002u Example ,)( ,)( ,1)( 2

321 xxfxxfxf

2x24321 Mfor basis a form } ,,, { uuuu


4dim 22 )(M x

3dim 3 )(R

3dim 2 )(P


0 such that

00


baba

x

W

V subspace 22xMW Find dim(W)


Definition: The dimension of a vector space V is the number of vectors in any basis of V

Example space vector a is 4RV

2 vectorsx41 all ofset the b][a,b,a,-W

V subspace W Find dim(W)


FACT: the solution set of Ax=0 is a subspace

Homogeneous Linear System

matrix an be Let mxnA (*) 0Ax(*) systemlinear theof ectorssolution v all ofset the define W

Consider the homogeneous linear system

nR

W

subspace

Example

014263

023142

055163

0A

)dim( Find c)

for basis a Find b)

spacesolution theFind a)

W

W

WConsider the homogeneous linear system

000000

041100

032021

E

How to find a basis for the solution space Wof the Homogeneous Linear System Ax=0

Example

014263

023142

055163

0A

)dim( Find c)

for basis a Find b)


W

W

W


000000

041100

032021

E

12

3

54

0matrix augmented thewrite A

0 formechelon -reduced theFind E

variablesleading :Identify

r, s, t, variablesFree :Set r, s, t, of in terms variablesleading :write

1 :Find 001 :set v, , t, sr 2 :Find 010 :set v, , t, sr 3 :Find 100 :set v, , t, sr

6

7 Wvvv for basis a is } , , , { 321

variablesFree :

How to find a basis for the solution space Wof the Homogeneous Linear System Ax=0

Example

03752

042310A

)dim( Find c)

for basis a Find b)


W

W

W


05310

0111101E

12

3

54

0matrix augmented thewrite A

0 formechelon -reduced theFind E

variablesleading :Identify

r, s, t, variablesFree :Set r, s, t, of in terms variablesleading :write

1 :Find 001 :set v, , t, sr 2 :Find 010 :set v, , t, sr 3 :Find 100 :set v, , t, sr

6

7 Wvvv for basis a is } , , , { 321

variablesFree :

:NOTE

dim( W ) = # of free variables

= # columns A - # of leading variables


ndim(V) d)

Vfor basis a is } ,,, { c)

Vspan } ,,, { b)


n21

n21

n21

vvv

vvv

vvv

Falseor True

22in dep lin. 60

01 ,

10

31 ,

02

45 ,

40

02 ,

03

21 a) xM

n

dependentlinearly } ,,,, { 1nn21 vvvv

3in dep lin.

6

2

1

,

7

3

1

,

4

0

3

,

5

1

2

b) R


ndim(V) d)

Vfor basis a is } ,,, { c)

Vspan } ,,, { b)


n21

n21

n21

vvv

vvv

vvv

Falseor True

22span ,02

45 ,

40

02 ,

03

21 a) xM

n

Vspan not does } ,,,, { 1nn21 vvvv

4span

2

9

1

3

,

1

5

4

2

,

3

1

2

1

b) R


Falseor True

22for basis 60

01 ,

10

31 ,

02

45 ,

40

02 ,

03

21 a) xM

n

3for basis

6

2

1

,

7

3

1

,

4

0

3

,

5

1

2

b) R

22for basis ,02

45 ,

40

02 ,

03

21 c) xM

4for basis

2

9

1

3

,

1

5

4

2

,

3

1

2

1

d) R

4for basis

6

2

4

2

,

2

9

1

3

,

1

5

4

2

,

3

1

2

1

e) R

4for basis

0

0

0

0

,

2

9

1

3

,

1

5

4

2

,

3

1

2

1

f) R


Vfor basis

} vectorsofet { stindependenlinearly

} vectorsofet { s

Vspan

} vectorsofet { s

n vectorsof#

2 conditions

out of 3

Falseor True

22for basis 10

31 ,

02

45 ,

40

02 ,

03

21 a) xM

3for basis

7

3

1

,

4

0

3

,

5

1

2

b) R

4for basis

2

0

1

0

,

2

9

1

3

,

1

5

4

2

,

3

1

2

1

d) R

Documents

Sec 4.2 + Sec 4.3 + Sec 4.4