Prep 2011 workshop Linear Algebra: Difference between revisions
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| Line 4: | Line 4: | ||
* Vectors | * Vectors | ||
** Geometric objects - lines and planes | ** Geometric objects - lines and planes | ||
** Dot product | ** Dot product and Vector Projections | ||
** Orthogonal decomposition | ** Orthogonal decomposition | ||
* Systems of equations and elimination | * Systems of equations and elimination | ||
| Line 14: | Line 13: | ||
** Matrix inverse | ** Matrix inverse | ||
** Matrix equations | ** Matrix equations | ||
** Determinant | ** Determinant and Cramer's Rule | ||
** Elementary matrices and LU Decomposition | ** Elementary matrices and LU Decomposition | ||
*** Note: There are determinant of elementary matrix questions in the NPL in Linear Algebra/Matrices/Determinants | *** Note: There are determinant of elementary matrix questions in the NPL in Linear Algebra/Matrices/Determinants | ||
Revision as of 17:48, 25 June 2011
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