Model Courses/ Linear Algebra

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
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
- Geometric examples
Linear transformations
- Definition of a linear transformation
- Matrix of a linear transformation
- Reflections, rotations, dilations and projections
- Inverse of a transformation
- Kernel, range, injection, surjection
Applications
- Graph theory: Adjacency matrix and Incidence Matrix
- Least squares
- Curve/surface fitting
- Mixture problems
- Simplex method
- Approximation of a function by a Fourier polynomial
Eigenvalues and eigenvectors
- Finding eigenvalues and eigenvectors
- Eigenspaces
- Diagonalization
- Symmetric matrices & Trace
- Quadratic forms
Inner product spaces and abstract vector spaces