Dr. Andrew G. has a Ph.D. from Caltech in environmental engineering science with a minor in numerical methods. In addition he has over 30 years experience as a practicing atmospheric scientist and dispersion modeler.
The concepts of Linear Algebra are at the heart of (1) numerical methods (used to develop and evaluate solution techniques that are used by computers to solve large numerical systems such as finite element analyses), (2) numerical solutions of overdetermined systems (i.e., "least squares" which are used in atmospheric receptor modeling techniques such as the Chemical Mass Balance model), and (3) statistical models (correlation and multiple regression).
Dr. Gray's research in air pollution includes the use of meteorological and dispersion computer models which involve numerical solutions of large linear (and non-linear) systems. Dr. Gray has also worked with (and helped in the development of) receptor models such as the Chemical Mass Balance Model and other related multivariate factor-analysis techniques. In addition, he has developed computer programs for the efficient (and stable) solution of linear systems (using such methods as Gram-Schmidt orthogonalization).
Topics in Linear Algebra include:
* Systems of Equations and Matrices - Systems of Equations, Row-Echelon Form, Reduced Row-Echelon Form, Gaussian Elimination, Gauss-Jordan Elimination, Matrices, Matrix Arithmetic, Transpose, Inverse Matrices, LU-Decompositions.
* Determinants - The Determinant Function, Properties of Determinants, Method of Cofactors, Determinants by Row Reduction, Cramer's Rule.
* Euclidean N-space - Vectors, Vector Arithmetic, Norm, Dot Product, Cross Product, Projections, Euclidean N-space, Linear Transformations.
* Vector Spaces - Vector Space Axioms, Subspaces, Span, Linear Independence, Linear Dependence, Basis, Dimension, Null Space, Row Space, Column Space, Inner Product Spaces, Orthogonal/Orthonormal Basis, Least Squares, QR-Decomposition, Orthogonal Matrices.
* Eigenvalues and Eigenvectors - Eigenvalues and Eigenvectors, Eigenspaces, Diagonalization, Jordan Canonical Form.