Prep 2011 workshop Linear Algebra

2011-06-23 !Preliminary Topic List

• Vectors
  • Geometric objects - lines and planes
  • Dot product
  • Projection
  • Orthogonal decomposition
• Systems of equations and elimination
  • Free variables
  • Consistency of solutions
  • Gaussian elimination
• Matrix operations and algebra
  • Matrix arithmetic
  • Matrix inverse
  • Matrix equations
  • Determinant
  • Elementary Matrices
  • LU
• Vector Space Preliminaries
  • Definition of a vector space
  • Euclidean vector spaces
  • linear combinations and span
  • linear independence
  • basis and orthogonal basis
  • coordinate vectors and change of basis
  • row space, column space, and null space
  • dimension
  • geometric examples
• Linear transformations
  • Matrix of a linear transformation
  • Reflections, rotations, dilations and projections
  • Inverse of a transformation
  • kernel, range, injection, surjection
• Applications
  • Adjacency matrix
  • Least squares
  • Curve/surface fitting
  • Mixture problems
  • Simplex method
  • Graph theory
• Eigenvalues and eigenvectors
  • Diagonalization
  • Symmetric matrices
• Inner product spaces and abstract vector spaces