This document discusses finite order vector spaces over finite fields. It defines vector spaces and properties such as subspaces, basis, and dimension. It then examines specific finite order vector spaces from 1 to 9 orders. For each order, it provides the Cayley diagram, properties, subspaces, basis, and dimension. The key points covered are that finite order vector spaces are isomorphic to Zn over Zn, and the number of non-isomorphic vector spaces increases with the order.

1. FiniteOrder
Vector
Spaces
By
Rajesh Bandari Yadav
Dr. K. Sarada
Finite order vectorspaces by Rajesh bandari yadav ,
K.Sarada is licensed under a Creative Commons
Attribution-NonCommercial 4.0 International License.

2. Finite order vector spaces
Contents
Introduction........................................................ 2
More about vector spaces..................................................3
Infinite order vector space ..................3
Finite order vector space................................ 4
2 SUBSPACES
Subspace .............................................................4
Properties of subspace
3
Linear combination & linear span .............................................5
Linear dependent & linear independent.......................................5
Basis & Dimension .........................................................5
4 Non-isomorphic finite order vector spaces & Properties
1-Order vector space.....................................6
2-Order vector space.....................................6
3-order vector space............................................. 6
4-Order vector space................................................................7
5-Order vector space..........................................................................8
7-Order vector space........................................................9
8-Order vector space.........................................9
9-order vector space.................................................... 11
11-Order vector space.....................................12
13-Order vector space......................................................13
16-Order vector space.................................................................13
17-Order vector space.........................................15
19-order vector space............................................. 16
4.14.Conclusion ..............................................................17
5. MATLAB programing for finding Non-Isomorphic finite order vectorspaces .............18
References.................................................................................... 21
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3. Finite order vector spaces
Introduction:
Def: Vector Space:
Let V be a non-empty set, (F, +, .) is a Field.
(V, +, .) is said to be vector space over the field F(denoted as V(F) ). If
i) (V, +) is an Abelian Group
• u+vϵV for all u,vϵV
• (u+v)+w=u+(v+w) for all u,v,wϵV
• 0ϵV such that 0+u=u+0=u for all uϵV
• For every uϵV their exist a unique vϵV such that u+v=v+u=0
• u+v=v+u for all u,vϵV
ii) a.uϵV for all aϵF,uϵV (Scalar multiplication)
iii) a.(u+v)=a.u+a.v for all aϵF, for all u,vϵV
iv) (a+b).v=a.v+b.v for all a,bϵF, for all vϵV
v) a.(b.v)=(a.b).v for all a,bϵF,for all vϵV
vi) for all vϵV such that 1.v=v where ‘1’ is unity in F. Ex:
R2={(x,y) ; x ,yϵR} is a vector space over the Filed (R,+ , .).
FINITE ORDER VECTOR SPACES
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4. Finite order vector spaces
R2 is a vector space over the field R
MORE ABOUT VECTOR SPACE:
i) Let (F, +, .) be a field.
➢ F (F) is a vector space
➢ Fn(F) is a vector space for n(≥1)ϵN.
➢ V=Mmxn[F] is vector space over the filed F.
➢ Let X be a non-empty set, V=FX is set of functions from X to F is a vector space over
the filed F.
ii) Let (F, +,.) and (E,+, .) are two fields such that ‘F’ is the sub field of ‘E’.
➢ E(F) is a vector space
➢ En(F) is a vector space for n(≥1)ϵN.
➢ V=Mmxn[E] is a vector space over the filed F.
➢ Let X be a non-empty set, V=EX is set functions from X to E is a Vector space over the
field F.
iii)
Vector Space
Infinite order
vector space
Finite
Dimensional
vector space
Infinite
Dimensional
vector space
Finite order
vector space
Finite
Dimensional
vector space
Infinite order Vector space:
A Vector space is said to be infinite order vector space if the vector space has
infinite elements.
Ex: Qn(Q) ,Rn(R), Cn(C)
Let (F, +, .) is the infinite order field
• V= Mmxn[F] is a v.s over F.
• V=FF is set of all function from F to F is a v.s over F.
• V=P[x] is the set of all polynomials whose coefficients belongs to F is a v.s over F
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5. Finite order vector spaces
𝑖=1
Finite order Vector space:
A Vector space is said to be finite order vector space if it has finite number of elements.
Examples:
• V={0} is a vector space (Null Space) of order 1 over any field F.
• V=(Z2 ,+2,x2) is a vector space of order 2 over the field (Z2 ,+2 ,x2) .
• Let ‘P’ be a prime
V=(ZP , +P ,xP) is a vector space over the filed (ZP ,+P ,xP) of order P.
• V=ZPXZPX…….XZP is a vector space over the filed (Zp, +p, xP) of orderPn.
‘n’ times
• V=Mmxn[ZP] is a vector space over the filed (ZP,+P, xP) of order Pmn.
• V=Pn[X] is set of all polynomials whose degree ≤n and coefficients are belongs ZP
is a vector space of order pn+1.
Note: Suppose V be a finite order vector space then |V|=P (or) Pn where P is a prime.
Sub Spaces:
Let V be vector space over the filed F, W be the any subset of V is said to be Subspace of V if
W is a vector space over the field F.
Properties of Subspace:
Let V be a vector space over the field F.
• W is the sub space of V iff a.u+b.vϵW for all a,bϵF, u,vϵV.
• W={0} is sub space of every vector space.
• If W is subspace of finite order vector space V then O(W)|O(V).
i.e If V is finite order vector space of order Pn then the proper subspace W of order is Pm
where m<n.
• W1, W2,………..Wk are subspaces of V then ⋂ 𝑘
• Union of two subspaces need not subspace of V.
𝑊𝑖 is subspace of V.
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6. Finite order vector spaces
Linear Combination & Linear span:
V be a vector space over the F,
Let S={v1, v2, …….vk} subset of V and vϵV .
Suppose ‘v’ can be written as v=a1v1+a2v2+------akvk then we say that ‘v’ is
linear combination of v1,v2,……..vk. where a1, a2, …..ak ϵF
Linearly independent & Linearly dependent:
A set S of vectors in a vector space is said to be linearly dependent if there exists a
vector ‘v’ in S such that ‘v’ belongs to the span of S-{v}.
S is linearly independent if S is NOT linearly dependent, that is, if ‘v’ does not belongs to
the span of S-{v} for each vϵS.
Basis & Dimension:
A subset S of Vector space V is said to be basis of V,if S is linearly independent and
L(S)=V.
Number of elements in basis is called dimension of V.
• Suppose V is a finite order vector space over the finite order filed ‘F ’with finite
dimension then o(V)=o(F)Dim(V)
• Let V is finite dimensional vector space of order Pn
i) The number distinct bases of V are
(Pn-1)(Pn-P)(Pn-P2)…….(Pn-Pn-1)
n!
ii)The number of k-dimensional subspaces of V is
(Pn-1)(Pn-1-1)………(Pn-k+1-1)
(Pk-1)(Pk-1-1)……(P-1)
• Suppose V be vector space over the fields (E,+,.) and (F,+,*) where F is the sub field of
E. then
Dim (V)=n over E
V E
Dim (V)=n.m over F Dim(E)=m over F
F
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7. Finite order vector spaces
• Two vector spaces have the same dimension over the same field then they are
isomorphic.
4.NON-ISOMORPHIC FINITE ORDER VECTOR SPACES & PROPERTIES
1-order vector space:
• V={0} is vector space of order 1 over any field F.
1. Subspaces: One and only one subspace.
2. Basis: empty set
3. Dimension: 0
2-Order vector space:
• V=(Z2 ,+2 ,X2) is a vector space over the field (Z2 ,+2 ,x2).
i. Properties:
• Every vector space of order 2 is isomorphic to (Z2 ,+2 ,X2).
• Number of non-isomorphic vector spaces of order 2 is only ONE.
ii. Subspaces:
• V has only trivial sub spaces
iii. Basis:
• B={1} is the basis of (Z2 ,+2 ,X2) over the field (Z2 ,+2 ,X2).
iv. Dimension: 1
3-Order vector space:
• V= (Z3 ,+3 ,X3) is vector space of order 3 over the field (Z3 ,+3 ,X3).
i. Caylay diagram:
ii. Properties:
• Every vector space of order 3 is isomorphic to V= (Z3, +3 ,X3).
• Number of non-isomorphic vector spaces of 3 is only ONE.
iii. Subspaces:
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8. Finite order vector spaces
• V has only trivial subspaces
iv. Basis:
• B1={1} ,B2={2} are two basis of V
v. Dimension: 1
4-Order vector space:
i)V1=(Z2XZ2 ,+, X) is a vector space of order 4 over the (Z2 ,+2, X2).
ii)Let (E,+,*) is a Field of order 4,
V2=(E,+ ,*) is a vector space of order 4 over the field (E.+,*).
i. Caylay diagram:
V= Z2XZ2
ii. Properties:
• Every vector space of order 4 is isomorphic to either V1 (or) V2 .
• Number of non-isomorphic vector space of order 4 is TWO.
iii. Subspaces:
a) Subspaces of V1=Z2XZ2 over Z2
V1=Z2XZ2
W 2={(0,0) , (0,1)} W3={(0,0) , (1,0) } W4={(0,0) , (1,1) }
W4= {(0, 0)}
Number total of subspaces are 5.
b) V2 has trivial subspaces.
c) Number of total subspaces are two.
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9. Finite order vector spaces
iv. Basis:
Basis of V1:
• B1={(1,0),(0,1)}
• B2={(1,0),(1,1)} are basis of V1=Z2XZ2
• B3={(0,1),(1,1)}
Basis of V2:
• Let a(≠0)ϵV2,the set B1={a} is the basis of V2 over the field(E,+,*)
v. Dimension:
• Dim(V1)=2
• Dim(V2)=1
5-order vector space:
V= (Z5, +5 ,X5) is vector space of order 5 over the field (Z5 ,+5 ,X5)
i. Caylay diagram:
ii. Properties:
• Every vector space of order 5 is isomorphic to V=(Z5 ,+5 ,X5).
• Number of non-isomorphic vector spaces of order 5 is ONE.
iii. Subspaces:
• V has only trivial subspaces.
iv. Basis:
• Let a(≠0)ϵV ,the set B={a} is the basis of V.
v. Dimension:
• Dim(V)=1
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10. Finite order vector spaces
4.6.7-Order vector space:
• V= (Z7, +7 ,X7) is vector space of order 7 over the field (Z7 ,+7 ,X7)
i. Caylay diagram:
ii. Properties:
• Every vector space of order 5 is isomorphic to V=(Z7 ,+7 ,X7).
• Number of non-isomorphic vector spaces of order 7 is ONE.
iii. Subspaces:
• V has only trivial subspaces.
iv. Basis:
• Let a(≠0)ϵV ,the set B={a} is the basis of V.
v. Dimension:
• Dim(V)=1
8-Order vector space:
• V1=(Z2XZ2XZ2 , +, X) is a vector space of order 8,over the field (Z2 ,+2 ,X2).
• Let (F,+,*) is filed of order 8,
V2=(F,+,*) is a vector space of order 8,over the field (F,+,*).
i. Caylay diagram: Page9

11. Finite order vector spaces
ii. Properties:
• Every 8-order vector space is isomorphic to one of V1 , V2 .
• Number of non-isomorphic vector spaces of order 8 is TWO.
iii. Subspaces:
• The sub spaces of V1=Z2XZ2XZ2 over the field Z2 are
8-Order subspaces is One
4-Order subspaces are 7.
W={(0,0,0),(0,1,1),(1,1,0),(1,0,1)
}
W={(0,0,0),(0,1,1),(0,1,0),(0,0,1)}
W={(0,0,0),(0,1,1),(1,1,1),(1,0,0)}
W={(0,0,0),(0,1,0),(1,1,0),(1,0,0)}
W={(0,0,0),(0,0,1),(1,1,0),(1,1,1)}
W={(0,0,0),(0,0,1),(1,1,0),(1,1,1)}
W={(0,0,0),(0,0,1),(1,0,1),(1,0,0)}
2-Order subspaces are
7. W={(0,0,0),(1,0,0)}
W={(0,0,0) ,(0,1 ,1)}
W={(0,0,0), (0,1,0)}
W={(0,0,0),(1,0,1)}
W={(0,0,0),(1,1,1)}
W={(0,0,0),(1,1,0)}
W={(0,0,0),(0,0,1)}
1-order subspace is
One W={(0,0,0)}
Hence total number of subspaces are 15 .
• The subspaces of V2 are only Two(Trivial)
iv. Basis:
• B={(1,0,0),(0,1,0),(0,0,1)} is basis of V1
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12. Finite order vector spaces
• B={a} where a(≠0)ϵV2 is the basis of V2.
v. Dimension:
• Dim(V1)=3
• Dim(V2)=1
9-Order vector space:
i)V1=(Z3XZ3 ,+, X) is a vector space of order 9 over the (Z3 ,+3, X3).
ii)Let (E,+,*) is a Field of order 9,
V2=(E,+ ,*) is a vector space of order 9 over the field (E.+,*)
i. Caylay diagram:
ii. Properties:
• Every vector space of order 9 is isomorphic to either V1 (or) V2 .
• Number of non-isomorphic vector space of order 9 is TWO.
iii. Subspaces:
a)Subspaces of V1=Z3XZ3 over Z3
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13. Finite order vector spaces
V1=Z3XZ3
W 2={(0,0) , (0,1) (0,2)} W3={(0,0) , (1,0),(2,0) } W4={(0,0) , (1,1),(2,2) } W5={(0,0),(2,1),(1,2)}
W6= {(0, 0)}
Number total of subspaces are
6. b)V2 has trivial subspaces.
c) Number of total subspaces are two.
iv. Basis:
Basis of V1:
• B1={(1,0),(0,1)}
• B2={(1,0),(1,1)} are basis of V1=Z3XZ3
• B3={(0,1),(1,1)}
• B4={(2,0),(0,2)}
• B5={(2,0),(0,1)}
• B6={(1,1),(0,2)}
Basis of V2:
• Let a(≠0)ϵV2,the set B1={a} is the basis of V2 over the field(E,+,*)
v. Dimension:
• Dim(V1)=2
• Dim(V2)=1
4.9.11-Order vector space:
• V= (Z11, +11 ,X11) is vector space of order 11 over the field (Z11 ,+11 ,X11)
i. Caylay diagram: Page12

14. Finite order vector spaces
ii. Properties:
• Every vector space of order 11 is isomorphic to V=(Z11 ,+11 ,X11).
• Number of non-isomorphic vector spaces of order 11 is ONE.
iii. Subspaces:
• V has only trivial subspaces.
iv. Basis:
• Let a(≠0)ϵV ,the set B={a} is the basis of V.
v. Dimension:
• Dim(V)=1
13-Order vector space:
• V= (Z13, +13 ,X13) is vector space of order 13 over the field (Z13 ,+13 ,X13)
i. Cayley digram:
ii. Properties:
• Every vector space of order 11 is isomorphic to V=(Z13 ,+13 ,X13).
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15. Finite order vector spaces
• Number of non-isomorphic vector spaces of order 11 is ONE.
iii. Subspaces:
• V has only trivial subspaces.
iv. Basis:
• Let a(≠0)ϵV ,the set B={a} is the basis of V.
v. Dimension:
• Dim(V)=1
16-Order Vector space:
• V1=(Z2XZ2XZ2XZ2 ,+ ,X) is a vector space over the field (Z2 ,+2 ,X2).
• Let (E,+,*) be a field of order 16,(F,+ ,*) is subfield of E of order
4. V2=(E,+,*) is a vector space of order over the filed (F,+,*) .
• Let (K,+,*) be a field of order 16,
V3=(K,+,*) is a vector space of order 16 over the filed (K,+,*).
i. Properties:
• Every vector space of order 8 is isomorphic to one of V1 ,V2,V3
• Number of non-isomorphic vector spaces of order 16 is THREE.
ii. Cayley diagram:
iii. Subspaces:
a)Sub spaces of V1:
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16. Finite order vector spaces
• 8-order subspaces of V1 are 15.
• 4-order subspaces of V1 are35.
• 2-order subspaces of V1 are 15.
• 1-order subspaces are
one Total subspaces of V1 are
66.
b)Sub spaces of V3: Only trivial subspaces.
iv. Dimension:
• Dim(V1)=4
• Dim(V2)=2
• Dim(V3)=1
17-Order Vector space:
• Every vector space of order 17 is isomorphic to V=(Z17 ,+17 ,X17).
i. Caylay diagram:
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17. Finite order vector spaces
ii. Properties:
• Every vector space of order 17 is isomorphic to V=(Z17 ,+17 ,X17).
• Number of non-isomorphic vector spaces of order 17 is ONE.
iii. Subspaces:
• V has only trivial subspaces.
iv. Basis:
• Let a(≠0)ϵV ,the set B={a} is the basis of V.
v. Dimension:
• Dim(V)=1
19-Order vector space:
• Every vector space of order 19 is isomorphic to V=(Z19 ,+19 ,X19).
i. Caylay diagram:
ii. Properties:
• Every vector space of order 19 is isomorphic to V=(Z19 ,+19 ,X19).
• Number of non-isomorphic vector spaces of order 19 is ONE.
iii. Subspaces:
• V has only trivial subspaces.
iv. Basis:
• Let a(≠0)ϵV ,the set B={a} is the basis of V.
v. Dimension:
• Dim(V)=1
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18. Finite order vector spaces
Counlcision :
Let V be a finite order vector space of order Pn
over the any finite order field F.
“The number of Non-Isomorphic Vector spaces of Pn
are Number of divisiors of n’’.
S.No Order of the Vector space Number of Non-isomorphic Vector spaces
1 1 1
2 2 1
3 3 1
4 4 2
5 5 1
6 7 1
7 8 2
8 9 2
9 11 1
10 13 1
11 16 3
12 17 1
13 19 1
14 23 1
15 25 2
16 27 2
17 29 1
18 31 1
19 32 2
20 37 1
21 41 1
22 43 1
23 47 1
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19. Finite order vector spaces
24 49 2
25 53 1
26 59 1
27 61 1
27 64 4
28 67 1
29 71 1
30 73 1
31 79 1
32 81 3
33 83 1
34 87 1
35 89 1
36 97 1
Number of non-isomorphic v.s of order below 100
5.MATLAB programing for finding Non-Isomorphic
finite order vectorspaces:
%CLASIFICATION OF FINITE ORDER VECTOR SPACES
disp('CLASIFICATION OF FINITE ORDER VECTOR SPACES')
k=1;
n=input('Give order of the vector space');
m=input('Give order of the field');
a=factor(n);b=factor(m);
if(isprime(n)&&isprime(m))
if(n==m)
fprintf('Their exist ONLY ONE NON-ISOMORPHIC Vector space over the
field')
fprintf('n')
disp(' S.No V.S Field Dim')
disp(' order order ')
disp(' ')
t1(:,1)=1;t1(:,2)=n;t1(:,3)=m;t1(:,4)=1;
disp(t1)
else
fprintf('Their does not exist vector space of order %i over the
field of order %i',n,m)
end
else
if(max(a)==min(a)&&max(b)==min(b))
x1=length(a);x2=length(b);
if(mod(n,m)==0 && mod(x1,x2)==0)
i=1:x1;
y=i(rem(x1,i)==0);
z=length(y);
fprintf('Their exist %i non-isomorphic vactor spaces of
order %i',z,n)
fprintf('n')
disp(' S.No V.S Field Dim')
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20. Finite order vector spaces
disp(' order order ')
disp(' ')
t2(:,1)=(1:z)';
t2(:,2)=(n*ones(1,z))';
t2(:,3)=(max(a).^y)';
t2(:,4)=rot90(y);
disp(t2)
else
fprintf('Their does not exist vector space of order %i over the
field of order %i ',n,m)
end
else
if(isprime(n)==0||isprime(m)==0)
fprintf('Their does not exist vector space of order %i over the
field of order %i',n,m)
end
end
end
s=input('n Do you want know number of linear operetor on a vectorspaces
,n press-1 n For Exit press-0');
if(s==1)
v1=input('Give vector space order');
F=input('Give order of the field');
E1=v1;
if(isprime(v1)&&isprime(F))
if(v1==F)
E2=v1-1;
fprintf('Their exist %i linear oparetos on given vectorspace ',E1)
fprintf('Their exist %i bijective linear oparetors on given vector
space',E2)
end
else
a1=factor(v1);a2=length(a1);a3=1;a4=(max(a1))^(a2*a2);
if(max(a1)==min(a1)&&max(a1)==F)
for i=0:(a2-1)
a3=a3*((max(a1))^a2-(max(a1))^i);
end
end
space',a4)
fprintf('Their exist %i linear opretors on given vector
fprintf('n Their exist %i bijective linear opretors on given
vector space',a3)
end
end
Output of the program
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21. Finite order vector spaces
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22. Finite order vector spaces
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23. Finite order vector spaces
References:
• A.Ramachandra Rao,P.Bhimasankaram-Linear Alebra
• Gilbert_Strang_Linear_Algebra_and_Its_Application( 4th edition).
• schaums's Linear algebra
• David C Lay -Linear Algebra And Its Applications
• I.N. Herstein-Topic in algebra
• Joseph Gallian -Contemporary Abstract Algebra 2009
• P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul-Basic Abstract Algebra-
Cambridge University Press (1994)
• Imeage sourse-Group Exploer software
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