Summary of "Vectors | Chapter 1, Essence of linear algebra"

Main Ideas and Concepts

Definition of Vectors:
- Vectors are fundamental building blocks in Linear Algebra.
- There are three perspectives on Vectors:
  - Physics Perspective: Vectors are arrows defined by length and direction.
  - Computer Science Perspective: Vectors are ordered lists of numbers (e.g., features like square footage and price).
  - Mathematician's Perspective: Vectors are generalized entities where Vector Addition and Scalar Multiplication are defined.
Geometric Representation:
- Vectors are often visualized as arrows originating from the coordinate system's origin (0,0) in 2D or (0,0,0) in 3D.
- Each vector corresponds to a specific coordinate pair (in 2D) or triplet (in 3D).
Coordinate System:
- In 2D, Vectors are represented as pairs of numbers (x, y), where:
  - The first number indicates movement along the x-axis.
  - The second number indicates movement along the y-axis.
- In 3D, Vectors are represented as triplets (x, y, z).
Vector Operations:
- Vector Addition:
  - To add two Vectors, place the tail of the second vector at the tip of the first vector and draw a new vector from the tail of the first to the tip of the second.
  - Numerically, Vector Addition involves adding corresponding components.
- Scalar Multiplication:
  - Multiplying a vector by a scalar stretches or shrinks the vector's length.
  - Each component of the vector is multiplied by the scalar.
Importance of Understanding Vectors:
- The ability to switch between the geometric and numerical representations of Vectors enhances comprehension and application in various fields like data analysis, Physics, and computer graphics.

Methodology/Instructions

Visualizing Vectors:
- Always think of Vectors as arrows in a coordinate system.
- For 2D Vectors, visualize them with their tail at the origin (0,0).
Adding Vectors:
- Use the tip-to-tail method:
  1. Position the second vector's tail at the tip of the first vector.
  2. Draw a new vector from the tail of the first to the tip of the second.
Multiplying Vectors by Scalars:
- To scale a vector, multiply each component by the scalar:
  - For example, if the vector is (x, y) and the scalar is k, the new vector will be (k*x, k*y).