Linear Algebra Concepts: Vector Coordinates and Basis Vectors

Overview

• This lecture covers vector coordinates, basis vectors, and introduces the concepts of span and linear dependence/independence.
• Importance of understanding how vector coordinates work and the multiple ways to describe them within linear algebra.

Vector Coordinates

• Vector Example: For a vector described by coordinates (3, -2):
  • Each coordinate acts as a scalar.
  • Scales unit vectors (i-hat and j-hat) in the x and y directions.
  • Results in a vector being described as a sum of scaled unit vectors.

Basis Vectors

• i-hat: Unit vector pointing right, direction of the x-axis.
• j-hat: Unit vector pointing up, direction of the y-axis.
• These vectors form the basis of the coordinate system.

Alternate Bases

• Basis vectors do not have to be i-hat and j-hat.
• Different basis vectors can form alternate but valid coordinate systems.
• All possible vectors are accessible through the scaling of these alternate basis vectors.

Linear Combination

• Definition: The sum of scaled vectors.
• Linear Combination Examples:
  • Fixing one scalar while altering the other traces a line.
  • Allowing both scalars to change freely can reach every 2D vector (unless vectors line up).
  • When vectors align, their linear combination lies on a single line.
  • Zero vectors result in a single point at the origin.

Span of Vectors

• Definition: Set of all possible vectors that can be reached through linear combinations of a given set of vectors.
• 2D Span:
  • Typically covers all of 2D space unless the vectors are collinear, limiting the span to a line.
• 3D Span:
  • Adding a third vector typically spans all 3D space unless the third vector is within the span of the first two.
  • If vectors are linearly dependent, the span doesn’t change.

Points vs. Vectors

• Vectors as Points:
  • Crowded space can simplify visualization by treating vectors as points.
  • Tip of a vector at the origin can be visualized as points in space.

Linear Independence and Dependence

• Linear Independence:
  • Vectors that each add a new dimension to the span are linearly independent.
• Linear Dependence:
  • Vectors that do not add a new dimension to the span are linearly dependent.

Basis of a Space

• Definition: A set of linearly independent vectors that span the space.
• Basis vectors are crucial for defining coordinate systems and transformations in space.

Next Steps

• Upcoming topics will include matrices and transforming space.