In this chapter, we study vector spaces and their basic properties and structures. We start by stating the definition and discussing examples of vector spaces. Next we introduce the notions of subspaces, linear dependence, bases, coordinates, and dimensionality. And then we consider dual spaces, direct sums, and quotient spaces. Finally, we cover normed vector spaces.

In this chapter, we consider linear mappings over vector spaces. We begin by stating the definition and discussing the structural properties of linear mappings. Then we introduce the notion of adjoint mappings and illustrate some of their applications. Next we focus on linear mappings from a vector space into itself and study a series of important concepts such as invariance and reducibility, eigenvalues and eigenvectors, projections, nilpotent mappings, and polynomials of linear mappings. Finally, we discuss the use of norms of linear mappings and present a few analytic applications.

In this chapter, everything is brought together to solve the MHD Riemann problem, the most general 1-D MHD problem one can solve semi-analytically. Non-linear waves are introduced in which the 1-D primitive equations are neither steady-state nor linearised. The fast and slow eigenkets are evaluated and their normalisation to account for the not-strictly hyperbolic nature of MHD is emphasised. A method to determine profiles of the primitive variables across slow and fast rarefaction fans is described, including Euler, switch-on, and switch-off fans. A strategy for solving the MHD Riemann problem follows, including use of a multi-variate secant root finder, sixth- order Runge–Kutta, and inverting a 5 × 5 Jacobian matrix with emphasis on characteristic degeneracy and matrix singularity. The chapter concludes with an explicit algorithm for an MHD Riemann solver including numerous examples using a solver developed by the author.

This chapter introduces the magnetic induction, B̅, to fluid dynamics. After a brief introduction establishing the ubiquity of magnetism in the universe, the ideal induction equation is derived from the idea of electromagnetic force balance and Faraday’s law. By proposing and proving the flux theorem, Alfvén’s theorem is proven to show that in an ideal MHD fluid, magnetic flux is conserved and frozen-in to the fluid. It is further shown how the introduction of Bti introduces the Lorentz force density to the momentum equation and the Poynting power density to the energy equation. Two variations of the equations of MHD are assembled, both involving the conservative variables. Finally, the vector potential, magnetic helicity, and magnetic topology are introduced in an optional section where the link to solar coronal flux loops is made.

By linearising the equations of HD developed in Chapter 1, the wave equation for the propagation of sound is derived. This is examined from two approaches: direct solution of the wave equation and examining normal modes to convert the problem to one of linear algebra. This introduces the very important concepts of eigenvalues (characteristic speeds) and eigenkets (right eigenvectors) along with the role they play in examining fluid dynamics in terms of waves. From the 1-D, non-linearised, steady-state equations, the Rankine–Hugoniot jump conditions are derived from which the conditions for tangential/contact discontinuities and shocks are developed. An optional section considers the phenomenon of bores and hydraulic jumps, while the last section introduces concepts such as streamlines and stream tubes culminating with Bernoulli’s theorem applied to an incompressible fluid, a subsonic compressible fluid, and a supersonic compressible fluid.

This chapter serves as the “practice chapter” for the main goal of Part I: solving the MHD Riemann problem. Lagrangian and Eulerian frames of reference are introduced from which the three Riemann invariants of HD are identified. Space-time diagrams are introduced as a useful visual and conceptual aid in understanding the role of characteristic paths through a continuum, which is in keeping with the text’s underlying approach of treating fluid dynamics as a form of wave mechanics. The Riemann problem for HD is defined and a method of characteristics is introduced whose main purpose is to understand qualitatively how the solution to the HD Riemann problem begins to unfold. In so doing, shocks and contact discontinuities are rediscovered and rarefaction fans are introduced. It is shown how examining the eigenkets leads to profiles of the primitive variables across a rarefaction fan which ultimately leads to a semi-analytic solution to the HD Riemann problem.

In this chapter, we present a few selected subjects that are important in applications as well but are not usually included in a standard linear algebra course. These subjects may serve as supplemental or extracurricular materials. The first subject is the Schur decomposition theorem, the second is about the classification of skew-symmetric bilinear forms, the third is the Perron–Frobenius theorem for positive matrices, and the fourth concerns the Markov or stochastic matrices.

A rich and important area for the applications of linear algebra is machine learning. In machine learning, one aims to achieve optimized or learned understanding of various kinds of real-world phenomena from data collected or observed, without real comprehension of the functioning mechanisms of such phenomena. These functioning mechanisms are often impossible or unpractical to grasp anyway. In this chapter, we present several introductory and fundamental problems in supervised machine learning including linear regression, data classification, and logistic regression and the mathematical and computational methods associated.