Learn & Review: Vectors | Essence of linear algebra, chapter 1

Summary of Vector Basics in Linear Algebra

This summary outlines the fundamental concept of a vector in linear algebra, exploring different perspectives and introducing the core operations of vector addition and scalar multiplication.

1. What is a Vector?

A vector is the foundational building block of linear algebra. There are three main perspectives:

• Physics Student Perspective:

  • Vectors are arrows in space.
  • Defined by their length and direction.
  • Their position can be moved without changing the vector itself.
  • Can be 2-dimensional (in a plane) or 3-dimensional (in space).

• Computer Science Student Perspective:

  • Vectors are ordered lists of numbers.
  • The number of elements in the list determines the vector's dimension.
  • The order of numbers is crucial.
  • Example: A house can be represented by a vector [square_footage, price].

• Mathematician's Perspective:

  • A generalization of the other two views.
  • A vector is anything where adding two vectors and multiplying a vector by a number are sensible operations.
  • This perspective is abstract and will be explored later.

2. Visualizing Vectors in Linear Algebra

For geometric understanding, it's helpful to visualize vectors as:

• Arrows originating from the origin of a coordinate system.
• This is a common convention in linear algebra, differing slightly from the physics view where vectors can be anywhere.

2.1. Two-Dimensional Coordinate System

• Consists of a horizontal x-axis and a vertical y-axis.
• The intersection is the origin, the starting point for vectors.
• Coordinates of a vector are a pair of numbers [x, y] that provide instructions to reach the vector's tip from its tail (at the origin).
  • The first number indicates movement along the x-axis (positive = right, negative = left).
  • The second number indicates movement parallel to the y-axis (positive = up, negative = down).
• Conventionally written vertically with square brackets:

  [ x ]
  [ y ]
  

• There's a one-to-one correspondence between pairs of numbers and 2D vectors.

2.2. Three-Dimensional Coordinate System

• Adds a third axis, the z-axis, perpendicular to both x and y axes.
• Vectors are associated with an ordered triplet of numbers [x, y, z].
  • The first number is movement along the x-axis.
  • The second is movement parallel to the y-axis.
  • The third is movement parallel to the z-axis.
• Every triplet corresponds to a unique 3D vector, and vice versa.

3. Fundamental Vector Operations

These operations are central to linear algebra.

3.1. Vector Addition

• Geometric Interpretation (Tip-to-Tail Method):
  1. Place the tail of the second vector at the tip of the first vector.
  2. The sum is a new vector drawn from the tail of the first vector to the tip of the second vector.
  • This method allows vectors to temporarily move away from the origin.
• Conceptual Understanding:
  • Each vector represents a movement (distance and direction).
  • Adding vectors is like performing sequential movements; the sum represents the overall net movement.
  • Analogous to adding numbers on a number line (e.g., 2 steps right + 5 steps right = 7 steps right).
• Numerical Implementation:
  • Add the corresponding components of the vectors.
  • If vector A = [a1, a2] and vector B = [b1, b2], then A + B = [a1 + b1, a2 + b2].
  • Example: [1, 2] + [3, -1] = [1+3, 2+(-1)] = [4, 1].

3.2. Multiplication by a Number (Scalar Multiplication)

• Geometric Interpretation:
  • Multiplying a vector by a number scales it.
  • A positive scalar stretches or shrinks the vector's length without changing its direction.
  • A negative scalar flips the vector's direction and then stretches or shrinks it.
• Terminology:
  • The number used for scaling is called a scalar.
  • The term "scalar" is often used interchangeably with "number" in linear algebra.
• Numerical Implementation:
  • Multiply each component of the vector by the scalar.
  • If vector V = [v1, v2] and scalar c, then c * V = [c*v1, c*v2].
  • Example: 2 * [1, 2] = [2*1, 2*2] = [2, 4].

4. The Power of Linear Algebra

• Linear algebra's strength lies in the ability to translate between different representations of vectors (geometric arrows vs. numerical lists).
• For Data Analysts: Provides a visual way to understand and conceptualize lists of numbers, clarifying patterns.
• For Physicists/Programmers: Offers a numerical language to describe and manipulate space, essential for applications like computer graphics.
• The core concepts of vector addition and scalar multiplication are fundamental to most topics in linear algebra.