Matricial norms, minimal matricial norms, vectorial norms and vectorial norms subordinate to matricial norms, which are respectively generalizations of matrix norms, minimal matrix norms, vector norms and vectorial norms subordinate to matrix norms, are defined and their various applications and properties are discussed. / Science, Faculty of / Mathematics, Department of / Graduate

Let U, V be two vector spaces of dimensions n and m, respectively, over an algebraically closed field F; let U⊗V be their tensor product; and let Rk(U⊗V) be the set of all rank k tensors in U⊗V, that is Rk(U⊗V) = {[formula omitted] are each linearly independent in U and V respectively}. We first obtain conditions on two vectors X and Y that they be members of a subspace H contained in Rk(U⊗V). In chapter 2, we restrict our consideration to the rank 2 case, and derive a characterization of subspaces contained in R2(U⊗V). We show that any such subspace must be one of three types, and we find the maximum dimension of each type. We also find the dimension of the intersection of two subspaces of different types. Finally, we show that any maximal subspace has a dimension which depends only on its type. / Science, Faculty of / Mathematics, Department of / Graduate

Let U⊗V be a tensor product space over an algebraically closed field F ; let dim U = m and dim V = n ; let T be a linear transformation on U⊗V such that T preserves rank two tensors. We show that T preserves rank one tensors and this enables us to characterize T for all values of m and n. / Science, Faculty of / Mathematics, Department of / Graduate

In this paper we present a theory of cylinder measures from the viewpoint of inverse systems of measure spaces. Specifically, we consider the problem of finding limits for the inverse system of measure spaces determined by a cylinder measure μ over a vector space X. For any subspace Ω of the algebraic dual X* such that (X,Ω) is a dual pair, we establish conditions on μ which ensure the existence of a limit measure on Ω . For any regular topology G on Ω, finer than the topology of pointwise convergence, we give a necessary and sufficient condition on μ for it to have a limit measure on Ω Radon with respect to G We introduce the concept of a weighted system in a locally convex space. When X is a Hausdorff, locally convex space, and Ω is the topological dual of X , we use this concept in deriving further conditions under which μ will have a limit measure on Ω Radon with respect to G. We apply our theory to the study of cylinder measures over Hilbertian spaces and ℓ(ρ)-spaces, obtaining significant extensions and clarifications of many previously known results. / Science, Faculty of / Mathematics, Department of / Graduate

The discovery of a new truth in the older fields of mathematics is a rare event. Here an investigator may hope at best to secure greater elegance in method or notation, or to extend known results by some process of generalization. It is our purpose to make a study of conic sections in the spirit of the above remark, using the symbolism developed by Josiah Williard Gibbs.

The purpose of this paper is to make a detailed study of vector spaces and a certain vector-like system.

A Dissertation submitted to the Department of Mathematics, Science and Technology Education In fulfilment of the requirement for the Degree of Master of Education (Mathematics Education) in the Faculty of Education at the University of Zululand, 2019. / Studies from interdisciplinary have noted positive correlation between spatial-visualization skills and mathematical problem solving. However, majority of these studies that interrogated this shared link between spatial-visualization and problem solving were carried in the urban settings only few interrogated rural settings. Also, studies have identified family social economic status (SES) which mainly described one’s geographical settlement to be one of the major effects on cognitive development. Thus, research finding from cognitive discipline revealed that students from poor SES background are less advantagous to cognitive activities (e.g., problem solving) compare to their counterpart. However, one of research achievements is providing evidence-based that cognitive skills can be enhanced through computer technology and spatial activities hence, the integration of several graphical tools such as: MATLAB, GeoGebra, and many other computer environments in mathematics education. These graphical tools are believed to enhance students’ conceptual and procedural knowledge in problem solving in mathematics areas such as: Euclidean geometry, multivariate calculus, and trigonometry which require more spatial skills in their problem solving. However, little has been researched on vector calculus even though vector calculus by its definition is accompanied by spatial reasoning. Students find it easy to evaluate a given vector integral using analytical techniques for integrations but struggle to visualize and transform it from one coordinate system to another. Objectives Based on the background, the current research employed the theoretical frameworks of Duval semiotic representation and the visual-analyser (VA) proposed by Zazkis et al., to analyse MATLAB instruction on rural-based pre-service teachers' spatial-visualisation skills and problem solving in vector calculus. The examination was guided by the analysis of the dynamic software MATLAB instruction on Spatial-Visualization, problem solving, and achievement in Vector Calculus. The three objectives were to 1) Analyse how rural-based v pre-service teachers apply their spatial-visualisation skills in problem solving in vector calculus. 2) To investigate the degree to which rural-based pre-service teachers’ spatial-visualisation skills correlate with their vector calculus achievement and 3) To assess how a dynamic software environment such as MATLAB influences rural-based pre-service teachers’ spatial-visualisation skills.

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