Prep 2011 workshop Linear Algebra

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Working page for the Linear Algebra group at PREP 2011

Preliminary Topic List - 2011-06-23

  • Vectors
    • Geometric objects - lines and planes
    • Dot product and Vector Projections
    • 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 and Cramer's Rule
    • 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

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