Topics of Linear Algebra

This chapter covers several topics of Linear Algebra, some of them not always treated in standard courses of mathematics for economic and financial analysis. Section 1.1 presents the axioms characterizing a linear space (or vector space) over a real field. Section 1.2 is concerned with basic notions on vectors in ℝ n $$\mathbb {R}^{n}$$ and with basic notions on topology for n -dimensional Euclidean spaces. Section 1.3 discusses basic topics on matrices, determinants and linear equations systems. Section 1.4 is concerned with linear functions or linear maps from ℝ n $$\mathbb {R}^{n}$$ to ℝ m . $$ \mathbb {R}^{m}.$$ Section 1.5 gives the basic notions on complex numbers, which are an essential tool to introduce the next section. Section 1.6 discusses eigenvalues and eigenvectors of square matrices: this subject is important in several areas of applied mathematics, such as the theory of quadratic forms (next section), the theory of nonnegative square matrices (Perron-Frobenius theorems), the analysis of several linear economic models of equilibrium and growth (Sect. 1.9), continuous and discrete dynamical systems (Chaps. 6 and 7 ). Quadratic forms (unconstrained and constrained), Perron-Frobenius theorems, an introduction to a simple Leontief input-output model and an introduction to a particular Sraffa economic model are the subjects of Sects. 1.7, 1.8, and 1.9.