The document defines several concepts related to groups and vector spaces: 1) It defines what a group is and lists the four properties a set must satisfy to be a group: closure, associativity, identity element, and inverses. 2) It provides examples of groups including real numbers, integers, and rational numbers under addition. 3) It proves that the set G={1,-1,i,-i} under multiplication is an abelian group by showing it satisfies the group properties. 4) It defines what a finite dimensional vector space is as a vector space that can be spanned by a finite set of vectors. 5) It defines linear dependence and independence of vectors in a vector space over

1. Group
Example
Theorem
Finite Dimensional V.S
Linear dependence & independence
Topic

2. Group
A non_empty set G is called group under binary operator
* if it hold the following conditions.
1)G obay Closure law.
∀𝑎, 𝑏 ∈ 𝐺
a*b∈G
2)G obay Associative law.
∀𝑎, 𝑏, 𝑐 ∈ 𝐺
a*(b*c)=(a*b)*c

3. Group
3) G contain an identity elements.
∀𝒂 ∈ 𝑮 & then their exist 𝒆 ∈ 𝑮
a*e=e*a=a
4) G contain inverse of each of its elements.
𝑎 ∈ 𝐺 then a-1∈ 𝐺
a*a-1=e

4. Examples
Note:-
set of real no. (R,+)
set of integer n0. (z,+)
Set of rational no. (Q,+)

5. Q:- Show that G={1,-1,I,-i} is an abelian group under ‘X’.
Solution:-
Q.Example
X 1 -1 i -i
1 1 -1 i -i
-1 -1 1 -i i
i i -i -1 1
-i -i i 1 -1

6. Q.Example
1) In table Closure law hold.
2) In table Associative law hold.
-1*(1*i) = (-1*1)*i
-1*(i) = (-1)*(i)
-i = -i
3) In table 1 is multiplication identity.
4) In table inverse of each elements exist.
(1)-1 =1 (-1)-1 = -1
(i) -1 =-i (-i)-1 =i

7. 5) The table is Symmetric along diagonal so G is an
abelian group.
Q.Example

8. Theorem
If S , T are subset of V. Then SCT <S> c <T>
Proof
Let x ∈<S>
Then x=𝜶 𝟏 𝒗 𝟏 + 𝜶 𝟐 𝒗 𝟐 + … … … … 𝜶 𝒎 𝒗 𝒎
with 𝒗𝒊′s∈S 𝜶 𝟏 ,s∈F
Since
SCT & 𝒗𝐢 ′s∈S
𝒗𝒊′s∈T
x=𝜶 𝟏 𝒗 𝟏 + 𝜶 𝟐 𝒗 𝟐 + … … … … 𝜶 𝒎 𝒗 𝒎 ∈<T>
Or
x ∈<T>
<S> ∈<T>

9. Muntha

10. Finite Dimensional V.S
A Vector space is said to be Finite Dimensional Vector
Space.
if we can find the finite set S such that
<S> = V
Spanning set Vector space

11. Let v be a vector space over field F. Then the vectors
x1,x2,x3,………………..,xn ∈ 𝑽 are said to be dependent
over F . if we can scalars 𝜶1, 𝜶2,……………, 𝜶n ∈F.
Such that not all Scalar are zero.
𝜶 𝟏x1 +𝜶 𝟐x2 + … … … … … . 𝜶nxn = 0
Linear Dependence

12. Linear Independence
Let v be a vector space over field F. Then the vectors
x1,x2,x3,………………..,xn ∈ 𝑽 are said to be independent over F .
if we can scalars 𝜶1, 𝜶2,……………, 𝜶n ∈F.Such that all Scalar are
zero.
𝜶 𝟏x1 +𝜶 𝟐x2 + … … … … … . 𝜶nxn = 0
𝜶 𝟏=0 , 𝜶 𝟐=0 , 𝜶 𝟑=0,________, 𝜶n =0

13. Theorem
Let U and w be a subspace of a vector space V over field F . The U∩W is also a
subspace of V.
PROOF
Let
𝜶 , 𝜷 ∈ F & x , y ∈ U ∩ W
Then
x , y ∈ U & x , y ∈W
since
U&W are subspace of V.
There for
𝜶x + 𝜷y ∈U & 𝜶x + 𝜷y ∈W
𝜶x + 𝜷y ∈U ∩ W
Hence U ∩ W is a subspace of V.