# Prep 2011 workshop Linear Algebra

From WeBWorK

# 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