The Three-Dimensional Real Vector Space ℝ³

The following sections are included:

• Vectorization of a Space: Affine Structure

• Coordinatization of a Space: ℝ³

• Changes of Coordinates: Affine Transformation (or Mapping)

• Lines in Space

• Planes in Space

• Affine and Barycentric Coordinates

• Linear Transformations (Operators)

  • Linear operators in the Cartesian coordinate system

  • Examples

  • Matrix representations of a linear operator in various bases

  • Linear transformations (operators)

  • Elementary matrices and matrix factorizations

  • Diagonal canonical form

  • Jordan canonical form

  • Rational canonical form

• Affine Transformations

  • Matrix representations

  • Examples

  • Affine invariants

  • Affine geometry

    • Affine independence and dependence (Sec. 3.6 revisited)

    • Affine subspaces

    • Affine coordinates and barycentric coordinates (Sec. 3.6 revisited)

    • Operations of affine subspaces

    • Dimension (or intersection) theorem

    • Relative positions of affine subspaces

    • Menelaus and Ceva theorems

    • Half-space, convex set, simplex and polyhedron

    • Affine mappings and affine transformations

    • Projectionalization of an affine space

  • Quadrics