Introductory Mathematics and Statistics for Islamic Finance, + Website

Chapter 6Linear Algebra

Islamic finance makes extensive use of linear algebra. In fact, linear algebra is one of the central disciplines in mathematics. It deals with vectors, matrices, and linear equations, and plays a major role in econometrics, linear programming, and the allocation of resources.

The chapter provides essential notions of linear algebra in the form of definitions and theorems. These notions are widely applied in Islamic finance and include operations using vectors and matrices, solutions of linear equations, computation of determinants and inverses of square matrices, and computation of characteristic equations, characteristic roots, and eigenvectors. It also addresses the notion of the stability of a linear system and Cholesky decomposition of a symmetric matrix.¹

Vectors

This section covers the addition of vectors, multiplication of vectors, vector space, linear combinations of vectors, linear dependence and linear independence of vectors, and bases of a vector space.

Scalars, vectors, and matrices are components of each other. A scalar is a one-dimensional vector, or a one-dimensional matrix; a vector is an ensemble of ordered scalars; and a matrix is a collection of vectors or scalars. In linear algebra, real numbers are called scalars. A scalar, generally speaking, is another name for a real number. Scalars are quantities that are fully described by a magnitude (or numerical value) alone. Examples of scalars are the following numbers: –1.5, 0, , , and ...