Objectives

In reading this chapter you will:

• learn about a major cause of infertility;

• reflect on the reasons for the shortage of donor gametes;

• consider what it would be like to have no idea who your parents are;

• reflect on ways to alleviate the gamete shortage;

• ask whether we should use eggs from aborted foetuses to alleviate the egg shortage;

• ask whether it is right to use artificial sperm to alleviate the sperm shortage;

• consider whether paying for reproductive resources is a morally unacceptable ‘commodification’ of the human body;

• reflect on artificial wombs and the extent to which they will relieve women of the burden of carrying babies to term.

A major cause of infertility is the inability to produce fertile gametes. Sincewomen have been having babies later this type of fertility has increased. Manymen have low sperm counts, or even no sperm at all and sperm counts quitegenerally seem to be decreasing. Sometimes the problems are not with gametesbut with difficulty in providing a womb hospitable to a developing foetus. Formany sub-fertile people, therefore, their chance of a child still depends on theirability to secure fertile gametes, and/or a hospitable womb. Securing these‘resources’ is not easy.

In this chapter we shall consider the ethical and social issues emerging from thedemand for and supply of gametes and of hospitable wombs. We shall start byconsidering gamete donation then turn to surrogacy.

We now turn our attention to special types of functions between vector spaces known as linear transformations. We will look at the matrix representations of linear transfor mations between Euclidean vector spaces, and discuss the concept of similarity of matrices. These ideas will then be employed to investigate change of basis and change of coordinates. This material provides the fundamental theoretical underpinning for the technique of diagonalisation, which has many applications, as we shall see later.

7.1 Linear transformations

A function from one vector space V to a vector space W is a rule which assigns to every vector v ∈ V a unique vector w ∈ W. If this function between vector spaces is linear, then it is known as a linear transformation, (or linear mapping or linear function).

Definition 7 .1 (Linear transformation) Suppose that V and W are vector spaces. A function T : V → W is linear if for all u, v ∈ V and all α ∈ ℝ:

1. T(u + v) = T(u) + T(v), and

2. T(αu) = αT(u).

A linear transformation is a linear function between vector spaces.

A linear transformation of a vector space V to itself, T : V → V is often known as a linear operator.

In this chapter, we look at orthogonal diagonalisation, a special form of diagonalisation for real symmetric matrices. This has some useful applications: to quadratic forms, in particular.

11.1 Orthogonal diagonalisation of symmetric matrices

Recall that a square matrix A = (ai j) is symmetric if AT = A. Equivalently, A is symmetric if ai j = aji for all i, j ; that is, if the entries in opposite positions relative to the main diagonal are equal. It turns out that symmetric matrices are always diagonalisable. They are, furthermore, diagonalisable in a special way.

11.1.1 Orthogonal diagonalisation

We knowwhat itmeans to diagonalise a square matrix A. Itmeans to find an invertible matrix P and a diagonal matrix D such that P-1 A P = D. If, in addition,we can find an orthogonal matrix P which diagonalises A, so that P-1 AP = PT A P = D, then this is orthogonal diagonalisation.

Definition 11.1 A matrix A is said to be orthogonally diagonalisable if there is an orthogonal matrix P such that PT AP = D where D is a diagonal matrix.

As P is orthogonal, PT = P-1, so PT A P = P-1 A P = D. The fact that A is diagonalisable means that the columns of P are a basis of ℝn consisting of eigenvectors of A (Theorem 8.22). The fact that A is orthogonally diagonalisable means that the columns of P are an orthonormal basis of ℝn consisting of an orthonormal set of eigenvectors of A (Theorem 10.21).

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.