Notes on Linear Algebra - Queen Mary University of London

Notes on Linear Algebra

Peter J. Cameron

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Preface

Linear algebra has two aspects. Abstractly, it is the study of vector spaces over fields, and their linear maps and bilinear forms. Concretely, it is matrix theory: matrices occur in all parts of mathematics and its applications, and everyone working in the mathematical sciences and related areas needs to be able to diagonalise a real symmetric matrix. So in a course of this kind, it is necessary to touch on both the abstract and the concrete aspects, though applications are not treated in detail.

On the theoretical side, we deal with vector spaces, linear maps, and bilinear forms. Vector spaces over a field K are particularly attractive algebraic objects, since each vector space is completely determined by a single number, its dimension (unlike groups, for example, whose structure is much more complicated). Linear maps are the structure-preserving maps or homomorphisms of vector spaces.

On the practical side, the subject is really about one thing: matrices. If we need to do some calculation with a linear map or a bilinear form, we must represent it by a matrix. As this suggests, matrices represent several different kinds of things. In each case, the representation is not unique, since we have the freedom to change bases in our vector spaces; so many different matrices represent the same object. This gives rise to several equivalence relations on the set of matrices, summarised in the following table:

Equivalence

Same linear map :V W

Similarity

Same linear map :V V

Congruence

Same bilinear form b on V

Orthogonal similarity Same self-adjoint : V V w.r.t. orthonormal basis

A = Q-1AP P, Q invertible

A = P-1AP P invertible

A = P AP P invertible

A = P-1AP P orthogonal

The power of linear algebra in practice stems from the fact that we can choose bases so as to simplify the form of the matrix representing the object in question. We will see several such "canonical form theorems" in the notes.

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These lecture notes correspond to the course Linear Algebra II, as given at Queen Mary, University of London, in the first sememster 2005?6.

The course description reads as follows:

This module is a mixture of abstract theory, with rigorous proofs, and concrete calculations with matrices. The abstract component builds on the notions of subspaces and linear maps to construct the theory of bilinear forms i.e. functions of two variables which are linear in each variable, dual spaces (which consist of linear mappings from the original space to the underlying field) and determinants. The concrete applications involve ways to reduce a matrix of some specific type (such as symmetric or skew-symmetric) to as near diagonal form as possible.

In other words, students on this course have met the basic concepts of linear algebra before. Of course, some revision is necessary, and I have tried to make the notes reasonably self-contained. If you are reading them without the benefit of a previous course on linear algebra, you will almost certainly have to do some work filling in the details of arguments which are outlined or skipped over here.

The notes for the prerequisite course, Linear Algebra I, by Dr Francis Wright, are currently available from

Alg I/

I have by-and-large kept to the notation of these notes. For example, a general field is called K, vectors are represented as column vectors, linear maps (apart from zero and the identity) are represented by Greek letters.

I have included in the appendices some extra-curricular applications of linear algebra, including some special determinants, the method for solving a cubic equation, the proof of the "Friendship Theorem" and the problem of deciding the winner of a football league, as well as some worked examples.

Peter J. Cameron September 5, 2008

Contents

1 Vector spaces

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1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Row and column vectors . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Subspaces and direct sums . . . . . . . . . . . . . . . . . . . . . 13

2 Matrices and determinants

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2.1 Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Row and column operations . . . . . . . . . . . . . . . . . . . . 16

2.3 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Calculating determinants . . . . . . . . . . . . . . . . . . . . . . 25

2.6 The Cayley?Hamilton Theorem . . . . . . . . . . . . . . . . . . 29

3 Linear maps between vector spaces

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3.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . 33

3.2 Representation by matrices . . . . . . . . . . . . . . . . . . . . . 35

3.3 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Canonical form revisited . . . . . . . . . . . . . . . . . . . . . . 39

4 Linear maps on a vector space

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4.1 Projections and direct sums . . . . . . . . . . . . . . . . . . . . . 41

4.2 Linear maps and matrices . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . 44

4.4 Diagonalisability . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5 Characteristic and minimal polynomials . . . . . . . . . . . . . . 48

4.6 Jordan form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.7 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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