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