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CHAPTER 4 Vector Spaces. Sec 4.2 + Sec 4.3 + Sec 4.4. Vector Space. Set:. 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 Addition:. Scalar Multiplication:. Set:. Vector Addition:. - PowerPoint PPT Presentation
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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:
Scalar Multiplication:
nnn
n
cvcv
cvcv
cv
1
111
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 nP
ndegrewith
ploynomial all ofset thenPSet:
)()())(( xgxfxgf Vector Addition:
Scalar Multiplication:
)())(( xcfxcf
Example space vector a is F
functions valuedreal ofset theFSet:
Vector Addition:
Scalar Multiplication:
)()())(( 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
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
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
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
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
CHAPTER 4 Vector Spaces
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
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
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
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:
Subspace
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 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
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
THM:
WvuWvu in then ,in , (1)
Example space vector a is 22xMV
0 such that
00
matrices 22 all ofset the
baba
x
W
VW
W subspace of VTwo conditions are satisfied
WcuWu in then ,in (2)
Example space vector a is 22xMV
01 matrices 22 all ofset the
baxW
Spanning setSec 4.2 + Sec 4.3 + Sec 4.4
CHAPTER 4 Vector Spaces
} ,,, { 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
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
} ,,, { 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
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
2321 Pfor basis a form } ,, { Hence, fff
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
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
2321 Pfor basis a form } ,, { Hence, fff
4dim 22 )(M x
3dim 3 )(R
3dim 2 )(P
Example space vector a is 22xMV
0 such that
00
matrices 22 all ofset the
baba
x
W
V subspace 22xMW Find dim(W)
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
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)
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
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)
spacesolution theFind a)
W
W
W
Consider the homogeneous linear system
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)
spacesolution theFind a)
W
W
W
Consider the homogeneous linear system
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
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
ndim(V) d)
Vfor basis a is } ,,, { c)
Vspan } ,,, { b)
tindependenlinearly } ,,, { a)
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
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
ndim(V) d)
Vfor basis a is } ,,, { c)
Vspan } ,,, { b)
tindependenlinearly } ,,, { a)
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
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
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
Sec 4.2 + Sec 4.3 + Sec 4.4CHAPTER 4 Vector Spaces
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