vector spaces a set v is called a vector space over a set k denoted v(k) if is an abelian group, is...
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Vector SpacesVector SpacesA set V is called a vector space over a set K denoted V(K) if ,V is an Abelian group,
,,K is a field,
and
For every element vV and K there exists an element .v V called the “scalar multiple of v by ” satisfying
• (i)
• (ii)
• (iii)
• (iv)
vuvuKVvu ...,,
vvvVvK ...,,
vvVvK ...,,
vv .1
Notation : 0 K denotes the additive identity under +, K, - denotes the inverse of under +
V0 denotes the identity under Vv v denotes the inverse of v under
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ExamplesExamplesExample 1 (Polynomials of degree n)
,,RKV = set of all polynomials of order n
The additive operation on vectors is defined as follows:
00111
11
011
1
011
1
baxbaxbaxba
bxbxbxb
axaxaxa
nnn
nnn
nn
nn
nn
nn
V(K) is a vector space
00000 1 xxx nn
Proof
v= 011
1 axaxaxa nn
nn
For 01
11 axaxaxav n
nn
n
Now vuuv by continuity of addition
on the real numbers
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Examples:1Examples:1
Proof continued
Closure is trivial
Associativity of follows from associativity ofnormal addition
011
1 axaxaxavR nn
nn
.,
therefore
uv
baxba
xbaxba
baxbaxbaxba
uv
nnn
nnn
nnn
nnn
.
.
.
0011
111
00111
11
Similarly for properties (ii)-(iv)
Hence, ,V is an Abelian group
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Examples:2Examples:2Example 2 (n dimensional vectors)
,,RK
RaaaV in ,,1
nn
nn
babauv
bbuaav
,,
,,,,,
11
11
then
Example 3 (Complex Numbers)
,,RK
RbabiaV ,
idbcauv
dicubiav
then
,
Example 4 (Matrices)
,,RK ,, ,mnMV
mnM , is the set of nm matrices
nM is the set of nn matrices
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Properties of Vector SpacesProperties of Vector SpacesTheorem
0.0 v
Proof
vv .00.0 Identity under +
vv .0.0 by axiom (ii) of vector spaces
But vv .00.0 Identity under
Therefore
vvv .00.0.0
0.0 v by the cancellation law for ,V
K
0
V
v
0
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Properties:2Properties:2Theorem
v. v.(i)
(ii) (-).( v) =.v
Proof
(i)
vv ..00 by previous theorem and inverse under +
Therefore,
vv ..0 by axiom (ii) of vector spaces
Also
= .( v)
0 (.v) v. by inverse under
Therefore
vv .. (.v) v.
v. (.v)
by the cancellation law for,V
Show v. v.
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Properties:3Properties:3Show v. = .( v)
.(v ( v)) 0. by inverse under
Therefore,
.v . v 0. by axiom (i) vector spaces
Now
Vv vv .0.00. by previous theorem
and axiom (iii)
0.0 v by above theorem
.v . v 0
Also
.v (.v) 0 by inverse under
Therefore,
.v . v= .v (.v)
. v= (.v) By the cancellation law for ,V
(ii) proof ??
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SubspacesSubspaces
Definition
Let WV such that W then W(K) is a subspace of V(K) if W is a vector space over K with the same definition of and scalar multiple as V
Clearly to show that W(K) is a subspace of V(K) we need only show that <W,> is a sub-group of <V, > and that
WuWuK .,
Characterisation Theorem
A non-empty subset W of V is a subspace of V iff .u vW for all K, u,vW
Proof
() trivial since .uW
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Subspaces:2Subspaces:2
Proof (continued)
() Taking =1 then u,vW
WvuWvu .1 by axiom (iv)
Taking =-1 then u,vW
Wvu .1 and by a previous theorem
(-1).u= (1.u)= u by axiom (iv)
Therefore u uW W0 by inverse under
Therefore K, uW taking 0v
gives .uW
Therefore, taking =-1 gives WuWu .1
(1.u)W uW
by a previous theorem
Hence, ,W is a subgroup and
WuWuK ., as required
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Examples of SubspacesExamples of Subspaces
Example 1
Let ,,RK
RaaaaW in ,,,, 320
RaaaV in ,,1
Then W(K) is a subspace of V(K)
W(K)
V(K)
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Examples of Subspaces:2Examples of Subspaces:2
Example 1 (continued)proof
(i) Clearly W and WV
(ii) For R and u,vW such that
naau ,,, 20 and nbbv ,,, 20
then
Wbabavu nn ,,,. 220
Example 2
RaaaV in ,,1
,,RK
RaaaaW in ,,,1 22
(i) Clearly W and WV
W0,,0,1,2 and 2R but
W 0,,0,2,40,,0,1,22 W(K) is not a subspace of V(K)
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Linear CombinationsLinear Combinations
Definition
If VvvS k ,,1 then
k
iiiv
1
where Ki
is a linear combination of S
1v
2v
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Linear Combinations:2Linear Combinations:2Theorem
For VvvS k ,,1 then
kiKvU i
k
iii ,,1,
1
is a subspace of V
Proof
U0 since Uvv k .0.0 1 and 0.0 iv
by a previous theorem and hence U
If UvuK ,, then kkvvu 11
and kkvvv 11 for some Kii ,
Then
kkkk
kk
kk
vvvv
vv
vvvu
1111
11
11
..
..
by axiom (i) kkkk vvvv 1111 ..
by axiom (iii) kkk vv .. 111 by axiom (ii)U
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Linear IndependenceLinear Independence
Definition
A subset kvv ,,1 of V is linearly independentiff
kivv ikk ,,1,0011 Otherwise if there exist one 0i such that
ikkvv 011
then kvv ,,1 are linearly dependent
Example 1
1,0,0,2,1,2,1,1,1,1,0,1
is linearly dependent over R since
0,0,01,0,002,1,21,1,11,0,1
Example 2
1,0,0,1,0,1 is linearly independent since
00,0
0,0,01,0,01,0,1
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Linear Dependence in Linear Dependence in MatricesMatrices
Theorem
KMaA nji ,If and jnjjj aaac ,,2,1 ,,,
nj ,,1 are the n column vectors of A then
ncc ,,1 is linearly dependent over K if and
only if 0det A
Proof
If ncc ,,1 is linearly dependent over K, then
there exists Kn ,,, 21 (not all zero)
such that0,,0,02211 nnccc
Without loss of generality assume 01 then
0,,0,01
21
21 n
n ccc
Hence, performing a column operation where
nn cc1
21
2
is added to column 1 gives
a matrix with zero first column. Hence, 0det A
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Matrices:2Matrices:2Proof (continued)
If 0det A then the system of equations
nixa j
n
jji ,,1:0
1,
has a non-trivial solution, nxx ,,1
But this is the same as saying that there exist
Kxx n ,,1 (not all zero) such that
0,,0,02211 nncxcxcx
By considering the transpose of A we obtain
Corollary
The n rows of a matrix KMA n are linearly
dependent if and only if 0det A
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BasisBasis
A set
Definition
kvvS ,,1 is a basis for V iff
(i) S is linearly independent over K
VkiKv i
k
iii
,,1,.1
(ii)
Condition (ii) means that S is a spanning set for V
Definition (Finitely Generated)
A vector space V is said to be finitely generated if it has a Basis S with a finite number of elements
S V
i
k
iiv
1
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Examples of BasisExamples of Basis
Let ,,RK
Example 1
RaaaaV i 321 ,,
Then 1,0,0,0,1,0,0,0,1 is a basis for V
Linear independence:
0
0,0,01,0,00,1,00,0,1
321
321
Spanning:
1,0,00,1,00,0,1,, 321321 aaaaaa
1,1,4,1,1,2,1,0,1 is also a basis for V
Proof ??
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Examples of Basis:2Examples of Basis:2
Example 2
RMV 2Let then 4321 ,,, vvvv is a basis
where
00
011v
00
102v
01
003v
10
004v
Example 3
RaaxaxaV i 012
2then
1,,2 xx is a basis
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DimensionDimension
Theorem Every Basis of a finitely generated vector space has the same number of elements.
Definition (Dimension)
The number of elements in a basis for a finitely generated vector space V is called the dimension of V and denoted dim V.
Examples
RaaxaxaV i 012
2then
1,,2 xx is a basis
dim(V) = 3 RMV 2Let then
is a basis and dim(V)=4
00
011v
00
102v
01
003v
10
004v