Linear Algebra. Many phenomena in the world have an underlying structure which follows basic algebraic rules. One of these structures is the vector space and linear algebra is the area of mathematics that has been developed to model phenomena that satisfy this structure. Competencies acquired in the successful study of linear algebra not only make it possible to study and understand the development of vector space models but also provide the foundation for the study of more advanced algebraic structures. The following competencies have been identified as essential for comparable preparation in this content area: Competency 1: Solving Systems of Linear Equations Competency 2: Matrix Arithmetic Competency 3: Determinants Competency 4: Vector Spaces Competency 5: Inner Product Spaces Competency 6: Eigentheory Competency 7: Linear Transformations See Appendix D: Competencies for Preparation in Linear Algebra.
Linear Algebra. 2. Differential Equations Students will not be penalized for not completing competencies in one or both of these areas of study, though exposure to these additional mathematical principles would greatly benefit a Math major transferring at the junior level. See Appendix A: Program-to-Program Articulation Model for Mathematics.
Linear Algebra. Many phenomena in the world have an underlying structure which follows basic algebraic rules. One of these structures is the vector space and linear algebra is the area of mathematics that has been developed to model phenomena that satisfy this structure. Competencies acquired in the successful study of linear algebra not only make it possible to study and understand the development of vector space models but also provide the foundation for the study of more advanced algebraic structures. The following competencies have been identified as essential for comparable preparation in this content area:
Linear Algebra. Multithreaded dense linear algebra in Parallel Colt is provided by JPlasma [117], which is our Java port of Parallel Linear Algebra for Scalable Multi-core Archi- tectures (PLASMA) [24]. An important matrix factorization for image processing applications is the singular value decomposition (SVD), but currently PLASMA does not have support for it. Therefore, Parallel Colt implements two sequential SVD algorithms. One is the original Colt version, which is essentially a slightly modified Jama [62] implementation, and the other is a divide-and-conquer rou- tine from JLAPACK (dgesdd). Note that our present use of the SVD in image processing is within a Krylov subspace method that enforces regularization on a (small) projected linear system; see [31]. Besides including JPlasma and JLAPACK in Parallel Colt, we have also added the following dense linear algebra operations: Kronecker product of 1D and 2D matrices (complex and real), Euclidean norm of 2D and 3D matrices computed as a norm of a vector obtained by stacking the columns of the matrix on top of one another, and backward and forward substitution algorithms for 2D real, upper and lower triangular matrices. Finally, we have implemented and included in Parallel Colt a Java version of the Concise Sparse Matrix Package (CSparse) [35], which we call CSparseJ [116]. Although CSparseJ is not multithreaded, it provides a set of matrix factorizations (LU, Cholesky and QR) that are much more efficient on sparse matrices than their dense equivalents. In the previous version of Parallel Colt, we used the same matrix factorization algorithms both for sparse and dense matrices (sparse matrices were converted to a dense form).
