In this short chapter, we aim to extend and consolidate what we have learned so far about systems of equations and matrices, and tie together many of the results of the previous chapters. We will intersperse an overview of the previous two chapters with two new concepts, the rank of a matrix and the range of a matrix.

This chapter will serve as a synthesis of what we have learned so far, in anticipation of a return to these topics later.

4.1 The rank of a matrix

4.1.1 The definition of rank

Any matrix A can be reduced to a matrix in reduced row echelon form by elementary row operations. You just have to follow the algorithm and you will obtain first a row-equivalent matrix which is in row echelon form, and then, continuing with the algorithm, a row-equivalent matrix in reduced row echelon form (see Section 3 .1.2). Another way to say this is:

• Any matrix A is row-equivalent to a matrix in reduced row echelon form.

There are several ways of defining the rank of a matrix, and we shall meet some other (more sophisticated) ways later. All are equivalent. We begin with the following definition:

Definition 4.1 (Rank of a matrix) The rank, rank(A), of a matrix A is the number of non-zero rows in a row echelon matrix obtained from A by elementary row operations.

Linear algebra is one of the core topics studied at university level by students on many different types of degree programme. Alongside calculus, it provides the framework for mathematical modelling in many diverse areas. This text sets out to introduce and explain linear algebra to students from any discipline. It covers all the material that would be expected to be in most first-year university courses in the subject, together with some more advanced material that would normally be taught later.

The book has drawn on our extensive experience over a number of years in teaching first- and second-year linear algebra to LSE undergraduates and in providing self-study material for students studying at a distance. This text represents our best effort at distilling from our experience what it is that we think works best in helping students not only to do linear algebra, but to understand it. We regard understanding as essential. ‘Understanding’ is not some fanciful intangible, to be dismissed because it does not constitute a ‘demonstrable learning outcome’: it is at the heart of what higher education (rather than merely more education) is about. Linear algebra is a coherent, and beautiful, part of mathematics: manipulation of matrices and vectors leads, with a dash of abstraction, to the underlying concepts of vector spaces and linear transformations, in which contexts the more mechanical, manipulative, aspects of the subject make sense.

A complex matrix is a matrix whose entries are complex numbers. A complex vector space is one for which the scalars are complex numbers. We shall see that many of the results we have established for real matrices and real vector spaces carry over immediately to complex ones, but there are also some significant differences.

In this chapter, we explore these similarities and differences. We look at eigenvalues and eigenvectors of a complex matrix and investigate unitary diagonalisation, the complex analogue of orthogonal diagonalisation. Certain results for real matrices and vector spaces (such as the result that the eigenvalues of a symmetric matrix are real) are easily seen as special cases of their complex counterparts.

We begin with a careful review of complex numbers.

Complex numbers

Consider the two quadratic polynomials, p(x) = x2 – 3x +2 and q(x) = x2 + x + 1. If you sketch the graph of p(x), you will find that the graph intersects the x axis at the two real solutions (or roots) of the equation p(x) = 0, and that the polynomial factorises into two linear factors: p(x) = x2 – 3x + 2 = (x - 1)(x - 2). Sketching the graph of q(x), you will find that it does not intersect the x axis. The equation q(x) = 0 has no solution in the real numbers, and it cannot be factorised over the reals. Such a polynomial is said to be irreducible. In order to solve this equation, we need to use complex numbers.

Chapter Overview

Soil has been described as the excited skin of the Earth and indeed it is, for like our own skin it is constantly changing as it takes in and gives up heat, water, chemicals, and organic matter. And like our skin, soil is a transition medium between two large spheres, and it shares traits of both, including air and water from above and rock and minerals from below. In this chapter we examine soil as a complex of systems, geomorphic, ecological, hydrologic, and biochemical. These systems are responsible for giving soil its basic form and composition, transforming what often begins as a chaotic mix of organic matter, particles of sediment, water, and other substances into an ordered whole. And since these systems are driven by a larger body of geographic systems, such as climate and hydrology, soils tend to vary correspondingly with these systems. That is, prairie soils are different than rainforest soils, which are different than desert soils, and so on.

Introduction

It was a late afternoon call from Jack Goodnoe. “Bill, what do you know about soils on Staten Island?” “Next to nothing,” I replied. “Why do you ask?” “Well, we've got a project there planning a new cemetery. It calls for thousands of burials, as well mausoleums, roads, waterlines, and landscaping.

Chapter Overview

Our planet is laced with shorelines, more than a million miles of them in all. They are infinitely varied in form, composition, and geographic character, but all have one trait in common: a capacity for unending change. This change comes from a wide variety of sources including earthquakes, tides, volcanoes, glaciers, land use, and hurricanes, but one system stands above all these as the premiere coastal change agent: the geomorphic system of waves and currents. This system operates almost everywhere all the time and is responsible for doing the lion's share of the work in eroding coastal land and transporting and depositing sediment. Our mission here is to understand how that system works, what drives it, and how it is capable of shaping the landforms of this celebrated environment. We also want to know how all this relates to humanity, because humans are particularly fond of the sea coast. Each year more and more people crowd into coastal lands throughout the world. We begin with a brief examination of the various systems that move sediment along the coast and then go on to the master system of wind waves, currents, wave erosion, and coastal landforms.

Introduction

It was a glorious summer morning. We loaded our little boat for a trip along the Lake Superior shore. “What are we looking for, anyway?” Jim asked. “Shoreline features,” I said without thinking.