Vector Spaces Lecture Notes

Introduction

• Presenter: Dr. Gajendra Purohit
• Topic: Vector Spaces (Linear Algebra series)
• Importance: Useful for competitive exams in higher mathematics
• Additional Resource: Another channel covering CSIR NET for life sciences, physics, mathematics, etc.

Prerequisites

• Understanding of Group Theory is essential:
  • Concepts: Group, Ring, Field
  • Recommended videos available in the playlist (link in the video)

Core Concepts

Internal and External Composition

• Internal Composition:
  • Involves a single set (V)
  • Operation on two vector elements must yield another vector (e.g., vector addition)
• Sets Involved:
  • V: Vector set
  • F: Field set
  • Mapping: f: V * F -> V

Properties of Vector Spaces

1. Closure:
   • The set V must be closed under addition.
2. Abelian:
   • Vector addition must be commutative.
3. Associativity:
   • Addition must be associative.
4. Identity Element:
   • The identity element with respect to addition is 0.
5. Inversibility:
   • For every element, there exists an inverse.

Scalar Multiplication

• The product of an element from F and a vector from V gives another vector.
• Closure under scalar multiplication is essential.

Examples of Vector Spaces

• C(R), C(Q), R(Q) are vector spaces.
• Q(Z) is NOT a vector space:
  • Reason: Z is not a field (no multiplicative inverse).

Proving Vector Spaces

• Example 1:
  • To check if a set is a vector space:
    1. Check if the sum of two vectors yields a vector in the set.
    2. If the result does not satisfy the equation (e.g., yields -36), it is not a vector space.
• N-tuples:
  • Proving that n-tuples of elements from any field F is a vector space:
    • Define V = Vn(F)
    • Check if (V, +) is an abelian group:
      • Associativity, commutativity, existence of identity (0), and inverses are verified.
    • Scalar multiplication properties must also be satisfied.

Operations and Properties

• For n-tuples (a1, a2, ..., an), addition is element-wise.
• Identity Element: 0 = (0, 0, ..., 0)
• Inverse Elements: If a tuple is (a1, a2, ..., an), the inverse is (-a1, -a2, ..., -an).
• Distributive Properties:
  • a*(alpha + beta) = aalpha + abeta
  • (a * b)alpha = a(b*alpha)

Conclusion

• The discussion on vector spaces will continue in future videos, focusing on subspaces and their properties.