Vector Spaces

Introduction

• Professor Dave discusses the concept of vector spaces (or linear spaces).
• Focus on generalizing operations like addition and scalar multiplication.

Notation

• Denote the vector space as V.
• An element of V is denoted as A ∈ V.

Properties of Elements in Vector Space

• Commutative Property: a + b = b + a
• Associative Property: (a + b) + c = a + (b + c)
• Existence of Zero Vector:
  • There exists a zero vector such that a + 0 = a.
• Existence of Additive Inverse:
  • For each element a, there exists -a in V such that a + (-a) = 0.
• Distributive Properties:
  • Scalar distribution: c(a + b) = ca + cb
  • Element distribution: (c + d)a = ca + da
• Multiplication by Scalar 1: 1 * a = a.*

Closure Property

• Closure requires two properties:
  1. For a ∈ V, c * a ∈ V for any scalar c.
  2. For any a, b ∈ V, a + b ∈ V.
• These properties determine whether V is a vector space.*

Examples of Closure

Set of Real Numbers (R)

• Multiplication: 5 * scalar results in a real number (e.g., 2 * 5 = 10).
• Addition: adding two real numbers results in another real number (e.g., 74 + (-10) = 64).
• R is closed; hence it is a vector space.

Real Vectors (Rn)

• Rn represents vectors of length n.
• Multiplying vectors by scalars and adding vectors yields vectors of the same length, confirming closure.

Matrices (Rm x n)

• Matrices of the same dimensions follow similar closure properties under addition and scalar multiplication.
• Thus, they also form a vector space.

Functions

• A vector space can consist of functions.
• Example: Linear polynomials of the form ax + b.
  • Closure under scalar multiplication and addition is maintained, confirming they form a vector space.

Failure of Closure Example

• Consider a set of 2D vectors where the second row is always 2.
• Adding two vectors from this set results in a vector with a second row value of 4 (which is not in the original set).
• Therefore, this set does not satisfy closure and is not a vector space.

Conclusion

• Understanding vector spaces and their properties is essential for future topics in the course.