Algebra Commons^™

The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

Electronic Theses, Projects, and Dissertations

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i² = 3, j² = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL₂(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …

Lecture 03: Hierarchically Low Rank Methods And Applications, David Keyes

Mathematical Sciences Spring Lecture Series

As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to “solve” a computational problem, which suggest that we have often been “oversolving” them at the …

Lecture 02: Tile Low-Rank Methods And Applications (W/Review), David Keyes

Mathematical Sciences Spring Lecture Series

As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to “solve” a computational problem, which suggest that we have often been “oversolving” them at the …

Lecture 11: The Road To Exascale And Legacy Software For Dense Linear Algebra, Jack Dongarra

Mathematical Sciences Spring Lecture Series

In this talk, we will look at the current state of high performance computing and look at the next stage of extreme computing. With extreme computing, there will be fundamental changes in the character of floating point arithmetic and data movement. In this talk, we will look at how extreme-scale computing has caused algorithm and software developers to change their way of thinking on implementing and program-specific applications.

An Invitation To Linear Algebra (2nd Edition), David N. Pham, Jonathon Funk, Wenjian Liu

Open Educational Resources

This is an OER textbook on linear algebra.

Algebra I Topics Using Geogebra, Matthew Rancourt

Masters Essays

No abstract provided.

An Invitation To Linear Algebra, David N. Pham, Jonathon Funk, Wenjian Liu

Open Educational Resources

This is an OER textbook on linear algebra.

Deblurring Images, Jamie Mcmullen

WWU Honors College Senior Projects

Let the matrix B be a blurred version of a sharp image represented by the matrix X. Given B, we would like to recover X.

To accomplish this, we construct linear models of the blurring process that produced B from X. The idea is that we could then reverse the blurring to reproduce the original image.

For example, if the blurred image satisfies

B = CXR^T

for some invertible matrices C and R, then we could recover X as

X = C-¹B(R^T)^-1.

However, the blurring model …

Webwork Problems For Linear Algebra, Hashim Saber, Beata Hebda

Mathematics Ancillary Materials

This set of problems for Linear Algebra in the open-source WeBWorK mathematics platform was created under a Round Eleven Mini-Grant for Ancillary Materials Creation. The problems were created for an implementation of the CC-BY Lyrix open textbook A First Course in Linear Algebra. Also included as an additional file are the selected and modified Lyryx Class Notes for the textbook.

Topics covered include:

• Linear Independence
• Linear Transformations
• Matrix of a Transformation
• Isomorphisms
• Eigenvalues and Eigenvectors
• Diagonalization
• Orthogonality

Linear Algebra (Ung), Hashim Saber, Beata Hebda, Piotr Hebda, Benkam Bobga

Mathematics Grants Collections

This Grants Collection for Linear Algebra was created under a Round Seven ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

Solving A System Of Linear Equations Using Ancient Chinese Methods, Mary Flagg

Linear Algebra

No abstract provided.

Normal Subgroups Of Wreath Product 3-Groups, Ryan Gopp

Williams Honors College, Honors Research Projects

Consider the regular wreath product group P of Z₉ with (Z₃ x Z₃). The problem of determining all normal subgroups of P that are contained in its base subgroup is equivalent to determining the subgroups of a certain matrix group M that are invariant under two particular endomorphisms of M. This thesis is a partial solution to the latter. We use concepts from linear algebra and group theory to find and count so-called doubly-invariant subgroups of M.

Linear Algebra, Daniel Scully

Math Faculty Publications

Table of Contents:

1. Systems of Linear Equations and Matrices

• Systems of Linear Equations
• Elementary Row Operations
• Row Reduction and Reduced Row-Echelon Form
• Solutions of Systems of Linear Equations
• Matrix Operations
• Matrix Inverses

2. Euclidean 2-Space and 3-Space

• Vectors in the Plane and in Space
• The Dot Product
• Cross Product
• Lines in Space
• Planes in Space

3. Determinants

• The Definition of Determinant
• Elementary Row Operations and the Determinant
• Elementary Matrices and the Determinant
• Applications of the Determinant

4. Vector Spaces and Subspaces

• Vector Spaces
• Subspaces
• Linear Dependence and Independence
• Basis and Dimension

5. Linear Transformations

• Definition of Linear Transformation
• The …

The Eigenvalues Of A Tridiagonal Matrix In Biogeography, Boris Igelnik, Daniel J. Simon

Electrical and Computer Engineering Faculty Publications

We derive the eigenvalues of a tridiagonal matrix with a special structure. A conjecture about the eigenvalues was presented in a previous paper, and here we prove the conjecture. The matrix structure that we consider has applications in biogeography theory.

Introduction To Linear Bialgebra, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic models and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerful algebraic tool that can yield itself to applications. With the recent introduction of bimatrices (2005) we have ventured in this book to introduce new concepts like linear bialgebra and Smarandache neutrosophic linear bialgebra and also give the applications of these algebraic …

An Invariance Property Of Common Statistical Tests, N. Rao Chaganty, A. K. Vaish

Mathematics & Statistics Faculty Publications

Let A be a symmetric matrix and B be a nonnegative definite (nnd) matrix. We obtain a characterization of the class of nnd solutions Σ for the matrix equation AΣA = B. We then use the characterization to obtain all possible covariance structures under which the distributions of many common test statistics remain invariant, that is, the distributions remain the same except for a scale factor. Applications include a complete characterization of covariance structures such that the chisquaredness and independence of quadratic forms in ANOVA problems is preserved. The basic matrix theoretic theorem itself is useful in other characterizing …

Introduction To Linear Algebra: Models, Methods, And Theory, Alan Tucker

Department of Applied Mathematics & Statistics Faculty Books

This book develops linear algebra around matrices. Vector spaces in the abstract are not considered, only vector spaces associated with matrices. This book puts problem solving and an intuitive treatment of theory first, with a proof-oriented approach intended to come in a second course, the same way that calculus is taught. The book's organization is straightforward: Chapter 1 has introductory linear models; Chapter 2 has the basics of matrix algebra; Chapter 3 develops different ways to solve a system of equations; Chapter 4 has applications, and Chapter 5 has vector-space theory associated with matrices and related topics such as pseudoinverses …

The Sharp Lipschitz-Constants For Feasible And Optimal-Solutions Of A Perturbed Linear Program, Wu Li

Mathematics & Statistics Faculty Publications

The purpose of this paper is to derive the sharp Lipschitz constants for the feasible solutions and optimal solutions of a linear program with respect to right-hand-side perturbations. The Lipschitz constants are given in terms of pseudoinverses of submatrices of the matrices involved and are proven to be sharp.

Linear Algebra By Analogy, Scott H. Hochwald

Scott H. Hochwald

No abstract provided.