Lecture Notes: Vectors in Linear Algebra

Introduction

• Instructor: Sarang Sane
• Topic: Introduction to vectors, part of the Maths 2 course in the online BSC program.

Content Overview

• Vectors and data
• Why vectors?
• Examples of vectors
• Visualization of vectors
• Vectors in physical contexts

---

Vectors and Data

• Data is often organized in tables.
• Examples of Data:
  • GDP sector-wise data from government website (2000-2001 to 2012-2013).
  • Team-wise batting averages of cricket players.
• Definition of Vectors:
  • Vectors can be thought of as lists.
  • Examples include rows and columns from tables.
  • E.g., Row vector from GDP table or column vector from batting averages.

---

Examples of Vectors

• Row and Column Vectors:
  • GDP example: A row vector could represent total GDP for a year.
  • Batting averages (e.g., Virat Kohli) as a column vector.
• Components of a Vector:
  • The number of components (e.g., a vector with 5 components).
• Column Matrix vs. Row Matrix:
  • A column vector or row vector can also be referred to as a column matrix or row matrix.

---

Why Vectors?

• Arithmetic Operations:
  • Vectors allow for arithmetic operations on lists, such as averaging.
  • Example: Average sectoral GDP across certain years by adding and dividing.
• Component-wise Operations:
  • Instead of doing operations entry-wise, vectors let us operate on lists more efficiently.

---

Illustrative Examples

• Buying Scenario:
  • Arun and Neela's rice and dal purchases.
• Grocery Stock Example:
  • Stock management using vectors to track sales and new stock arrivals.
• Vector Addition:
  • Adding vectors component-wise for total stock.
• Scalar Multiplication:
  • Example of doubling buyer A's purchases across two days.

---

Visualization of Vectors

• In R2 (2D space):
  • Points represented as vectors (e.g., (a, b)).
  • Vectors can be visualized as arrows from the origin to a point (e.g., (1, 2) as an arrow from (0, 0) to (1, 2)).
• Vector Addition in R2:
  • Two vectors can be added using head-to-tail method or parallelogram law.

---

Vectors in Rn

• Definition:
  • Vectors in Rn are lists with n real entries.
  • Similarity between vectors and points in terms of representation.
• Importance of Differentiation:
  • Vectors have operations such as addition and scaling, while points do not.

---

Vectors in Physical Contexts

• Vectors have both magnitude and direction (important in physics).
• Examples:
  • Velocity, acceleration, and force are vector quantities.
• Real-life Applications:
  • Example of a plane's movement affected by wind.

---

Conclusion

• Focus on vectors in the context of data for this course.
• Vectors should be treated as lists for algebraic operations (coordinate/entry-wise).
• Future studies will introduce more geometric concepts with coordinates.

---

Further Remarks

• Remember to differentiate when studying vectors algebraically vs. geometrically.
• Scalar multiplication and operations on vectors are crucial for understanding applications in data.

---

Thank You

• Hope this lecture provided clarity on the concept of vectors.