Vector Space in Linear Algebra : - Definition, Properties, Examples and Applications.

 Introduction :-

A Vector Space is a set of objects called vectors together with two operations:

1. Vector Addition

2. Scalar Multiplication

These operations must satisfy certain mathematical rules known as Vector Space Axioms.

A vector space allows us to perform addition and multiplication consistently while preserving algebraic structure.

Vector Space :- 

Let (F, +, .) be a field and V be non empty set. The elements of F are called scalars and the elements of V are called Vectors, Then V is a vector space over field F if -

1. u + v ∈ V , ∀ u, v ∈ V

2. α.u ∈ V , where α ∈ F , u ∈ V.

3. u + (v + w) = (u + v) + w

4. V has a zero vector in V such that u + 0 = u. (Additive Identity)

5. For every u in V there is a vector -u in V such that u + (-u) = 0. (Additive Inverse)

6. u + v = v + u

7. α(u + v) = αu + αv

8. (α + β).u = αu + βu

9. α(βu) = (αβ)u

10. 1.u = u , where 1 is the unit element of the field F.

नोट :- Vector Space मे V और F दोनो Field होते है !

👉 Vector Space को check करने के लिए सभी Axioms Check करने की जरूरत नही है , केवल ये Axioms Check करो यदि कोई भी conditions Fail होती है तो वो Vector Space नही बनेगा  , Vector Space is denoted by V(F).

                                 

                                            

                                          EXAMPLES OF VECTOR SPACE

Example 1. The set of all n- tuples of elements in K^n is a vector space over the field K.

Example:- 2 Proof that P(t) (Set of all polynomials) where coefficients belongs to a field K is a vector space over the field K.

Note :- Same Degree के polynomial कभी भी vector space नही बनाते 

Ex:- Let p(x) , q(x) ∈ V, where p(x) = x⁴ + 1 and q(x) = -x⁴ + 1 but p(x) + q(x) = 2 which does not belongs to V. Hence it is not a vector Space.

Example :- 3 Proof that the set of all m x n matrices whose entries belongs to field K is a vector space over the field K.

Example :- 4 Proof that the set of all functions F(X) in K is a vector space over K.

Conclusion

A Vector Space is a mathematical structure where vectors can be added together and multiplied by scalars while satisfying ten fundamental axioms.

Mastering this concept makes advanced topics such as Linear Transformations, Basis, Dimension, Eigenvalues, and Inner Product Spaces much easier to understand.

Play With Math – Learn Mathematics Conceptually, Not Memorically!