Notes on Vector Spaces Lecture

Introduction

• Continuation of the previous video on vector spaces.
• Focus on properties of vector spaces and the cancellation law of vector addition.

Cancellation Law of Vector Addition

• Definition: If (v_1 + v_3 = v_2 + v_3), then (v_1 = v_2).

Example in ( , ext{R}^n )

• Consider vectors in ( , ext{R}^3 ):
  1. For vectors (x = (x_1, x_2, x_3)) and (z = (z_1, z_2, z_3)) in ( , ext{R}^3 ):
     • If (x_1 + y_1 = z_1 + y_1), then subtract (y_1): (x_1 = z_1).
     • This extends to all components (i = 1, 2, 3).

General Vector Spaces

• The same reasoning applies to any vector space due to the axioms governing vector addition.
• Use of the zero vector (v_3') as part of the axioms to show that (v_1 = v_2).

Corollaries of Cancellation Law

1. Uniqueness of Zero Vector: There cannot be multiple vectors satisfying the zero vector property.
2. Uniqueness of Negative Vectors: For each vector (v), there exists a unique vector (-v).

Proof of Uniqueness

• If (w) also satisfies (v + w = v), then:
  • Show (w = 0) by cancellation.

Important Properties of Vector Spaces

• For any vector (v) and scalar (c):
  • (0 \times v = 0)
  • (-c \times v = - (c \times v))
  • (c \times 0 = 0) (left as an exercise).

Real-Life Example: Grocery Store

• Vectors can represent inventory:
  • Items: rice, dal, oil, biscuits, soap.
  • Each column in a table represents a vector in an abstract vector space.
  • Addition and scalar multiplication are defined coordinate-wise.

Understanding Negative Quantities

• Negative quantities correspond to demand (e.g., -2 kgs of rice = demand for 2 kgs).
• Use real numbers to represent quantities flexibly.

Geometric Interpretation of Vector Spaces

• Consider a plane parallel to the xy-plane for vector operations:
  • Scalar Multiplication: Project point onto the xy-plane, scale, and project back.
  • Addition: Project both points onto the xy-plane, add them using parallelogram law, and project the result back.

Summary

• Explored properties of vector spaces, including zero and negative vectors.
• Real-world application in grocery stock management.
• Geometric interpretation of vector operations.

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Final Thoughts

• Understanding vector spaces requires combining algebraic properties with geometric intuition.