Lecture on Vector Coordinates, Basis and Span

May 31, 2024

Lecture on Vector Coordinates, Basis and Span

Introduction

  • Review on vector addition and scalar multiplication
  • Discussion of vector coordinates in the context of linear algebra

Vector Coordinates

  • Pairs of numbers signify vectors (e.g., (3, -2))
  • Coordinates as scalars that stretch/squish vectors

Basis Vectors

  • Special vectors in the x-y system:
    • i-hat: unit vector in the x direction (1, 0)
    • j-hat: unit vector in the y direction (0, 1)
  • Coordinates can scale these basis vectors:
    • x-coordinate scales i-hat
    • y-coordinate scales j-hat
  • Vector described as a sum of scaled basis vectors
  • Basis: The set of vectors that the coordinates scale

Alternative Basis

  • New coordinate systems can be generated with different basis vectors
  • By choosing different basis vectors, the numerical description of vectors changes
  • Challenge to think about vectors generated by different pairs of basis vectors

Linear Combinations

  • Adding scaled vectors from chosen scalars
  • The term 'linear' refers to the resulting vector drawing a straight line when one scalar is fixed
  • Linear combination of two vectors can:
    1. Reach every possible 2D vector (if vectors don’t line up)
    2. Be limited to a line (if vectors line up)
    3. Be at origin (if both vectors are zero)
  • Span: The set of all possible vectors from linear combinations of a given pair of vectors

Vector Points

  • Vectors often represented as points for simplicity
  • Crowding avoided by representing collections of vectors as points
  • Single vectors = arrows; collections of vectors = points

Span in 3D Space

  • Two vectors in 3D space span a flat sheet (plane) through the origin
  • Adding a third vector to span:
    • If third vector within the span of first two, no new vectors reached
    • Generally, a third vector unlocks access to full 3D space

Linear Dependence and Independence

  • Linearly dependent: A vector can be expressed as a linear combination of others
  • Linearly independent: Each vector adds a new dimension to the span

Puzzle and Conclusion

  • Technical definition of basis: Set of linearly independent vectors that span a space
  • Challenge to think about why this definition makes sense
  • Next topic preview: Matrices and transforming space