Prep 2011 workshop Linear Algebra

Working page for the Linear Algebra group at PREP 2011

Preliminary Topic List - 2011-06-23

• Vectors
  • Geometric objects - lines and planes
  • Dot product
  • Projection
  • Orthogonal decomposition
• Systems of equations and elimination
  • Row operations and Row Echelon Form
  • Gaussian elimination (Free variables & Consistency of solutions)
• Matrix operations and algebra
  • Matrix arithmetic
  • Matrix inverse
  • Matrix equations
  • Determinant
  • Elementary matrices and LU Decomposition
    • Note: There are determinant of elementary matrix questions in the NPL in Linear Algebra/Matrices/Determinants
• Vector space preliminaries
  • Definition of a vector space and subspaces
  • 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 and rank
    • Note: There are some "rank" problems in the NPL in Linear Algebra/Matrices/Matrix Operations
  • 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
  • Approximation of a function by a Fourier polynomial
• Eigenvalues and eigenvectors
  • Finding eigenvalues and eigenvectors
  • Eigenspaces
  • Diagonalization
  • Symmetric matrices & Trace
    • Note: There are some "trace" problems in the NPL in Linear Algebra/Matrices/Matrix Operations
  • Quadratic forms
• Inner product spaces and abstract vector spaces