Chapter 4, Section 1: Vector Spaces and Subspaces

Introduction

• The course material is becoming more abstract and challenging.
• Students may need additional resources such as textbooks or supplementary videos to fully grasp the concepts.

Definition of a Vector Space

• A vector space is a non-empty set of vectors with two operations: addition and multiplication.
• Must satisfy specific axioms for all vectors ( u, v, ) and ( w ) in the set and scalars ( c ) and ( d ) in the real numbers.

Axioms of Vector Spaces

1. Closure under Addition: Sum of two vectors in the set is also in the set.
2. Commutative Property: ( u + v = v + u ).
3. Associative Property: ( (u + v) + w = u + (v + w) ).
4. Existence of Zero Vector: A zero vector exists such that ( 0 + v = v ).
5. Existence of Additive Inverse: For every vector ( v ), there’s a vector (-v) such that ( v + (-v) = 0 ).
6. Closure under Scalar Multiplication: Scalar multiple of a vector is also in the set.
7. Distributive Properties:
   • ( c(u + v) = cu + cv )
   • ( (c + d)v = cv + dv )
8. Associative Scalar Multiplication: ( c(dv) = (cd)v ).
9. Identity for Scalar Multiplication: ( 1v = v ).

Common Issues with Vector Spaces

• Key properties often checked first:
  • Closure under addition.
  • Existence of zero vector.
  • Closure under scalar multiplication.

Examples of Vector Spaces

Example 1: Quadrant IV Vectors

• Set ( V ) of vectors ((x, y)) where ( x \geq 0 ) and ( y \leq 0 ).
• Check against axioms:
  • Closure under addition: May fail if sum is not in Quadrant IV.
  • Zero vector exists because ( x = 0, y = 0 ).
  • Closure under scalar multiplication fails for negative scalars (e.g., (-3v)).
• Conclusion: Not a vector space.

Example 2: Positive Quadrant

• Set of vectors where both ( x, y > 0 ).
• Zero vector is not included as it doesn't satisfy ( x, y > 0 ).
• Conclusion: Not a vector space.

Example 3: XY Plane Regions

• Vector set where ( xy \leq 0 ).
• Zero vector included as ( 0 \leq 0 ).
• Closure under addition may fail based on vector selection.
• Conclusion: Not a vector space.

Example 4: Polynomials

• Polynomials form a vector space.
• Polynomials can be seen as linear combinations.
• Verify axioms:
  • Closure under addition: Adding polynomials results in a polynomial.
  • Zero polynomial exists.
  • Closure under scalar multiplication is satisfied.
• Conclusion: Polynomials satisfy all axioms of vector spaces.

Conclusion

• Understanding vector spaces involves verifying each axiom for a given set.
• Be aware of the challenges, especially with abstract examples.
• Practice with different examples to strengthen understanding.