Lecture on Vector Spaces by Dr. Gajendra Purohit

Introduction

• Dr. Gajendra Purohit's YouTube Channel: Engg Mathematics & BSc content.
• Useful for competitive exams with higher mathematics.
• New topic series: Vector Spaces in Linear Algebra.

Pre-requisites

• Understanding of Group Theory:
  • Concepts of Group, Ring, and Field.
  • Internal and External composition.
  • Videos available on Dr. Purohit's channel.

Vector Spaces Basics

• Vector Spaces are fundamental in Linear Algebra.
• Important terms:
  • V: Set of vectors.
  • F: Field.
  • Internal Composition: Operation within set V should return a vector.
  • External Composition: Mapping involves both elements from F and V.

Properties of Vector Spaces

1. Closed under Addition: Operation on two elements should return a vector.
2. Abelian Group (commutative): Addition should be commutative.
3. Associativity: Vector addition should be associative.
4. Identity Element: Existence of an additive identity (0).
5. Inversibility: Every vector should have an additive inverse.
6. Scalar Multiplication Closure: Operation with a scalar from field F and vector from V should yield another vector in V.

Examples & Non-examples

• Q(z): Not a vector space because Z is not a field, lacks multiplicative inverses.
• C(R), C(Q), R(Q): All valid vector spaces as they satisfy the properties.

Proving a Vector Space

• To prove a set is a vector space, follow these steps:
  1. Verify Closure: Sum of vectors should result in a vector within the set.
  2. Check Vector Addition: Satisfy commutativity, associativity, identity, and inversibility.
  3. Scalar Multiplication: Product of any scalar from field F and a vector from set V should land in V.

Worked Example:

1. N-tuples Elements (a1, a2, ... , an): Prove Vn(F) as a vector space.
2. Abelian Group: Verify commutativity, associativity, and identity existence.
3. Scalar Multiplication:

• Prove distributive properties over addition and scalar multiplication.
• Example shown with elements alpha, beta, and scalar a.

Conclusion

• Demonstrated properties needed for vector spaces.
• Provided examples and non-examples to clarify concepts.
• Upcoming videos will cover more on vector spaces, subspaces, and their properties.