1. Vectors.- 1. Notation.- 2. Definition of vectors.- 3. Addition of vectors.- 4. Multiplication by a scalar.- 5. Geometrical interpretation.- 6-7. Linear dependence of vectors.- 8. A basis for the set of n-vectors.- 9. The vector space spanned hy a finite set of vectors.- Exercises on chapter I.- 2. The Solution of a System of Equations: the Regular Case.- 1-2. Regular systems. Notations and statement of results.- 3-4. Elementary operations on systems.- 5-7. Proof of the Main Theorem.- 8-9. Illustrations to the Main Theorem.- 10. The linear dependence of n + 1 vectors in n dimensions.- 11. The construction of a basis.- Exercises on chapter II.- 3. Matrices.- 1-2. Definition of a matrix.- 3. The effect of matrices on vectors.- 4. Equality of matrices.- 5. Addition of matrices and multiplication hy a scalar.- 6. Multiplication of square matrices.- 7. The zero-matrix and the unit-matrix.- 8. Multiplication of matrices of any shape.- 9. The transpose of a matrix.- Exercises on chapter III.- The Solution of a System of Equations: the General Case.- 1-2. The general system and the associated homogeneous system.- 3. The inverse of a regular matrix.- 4. Computation of the inverse matrix.- 5. Application to the solution of regular systems.- 6. The rank of a matrix.- 7. The solution of homogeneous systems.- 8. Illustrations.- 9. The solution of general systems.- 10. Illustrations.- 11. Geometrical interpretation.- Exercises on chapter IV.- 5. Determinants.- 1. Motivation.- 2. The 2-dimensional case.- 3. The 3-dimensional case.- 4. The rule of signs in the 3-dimensional case.- 5. Permutations.- 6. The Kronecker ?-symbol.- 7. determinant of an n × n matrix.- 8. Co factor s and expansions.- 9. Properties of determinants.- 10. An expression for the cofactors.- 11. Evaluation of determinants.- 12. A formula for the inverse matrix.- 13. Cramer's Rule.- 14. The Multiplication Theorem.- 15. A determinantal criterion for the linear dependence of vectors.- 16. A determinantal expression for the rank of a matrix.- Exercises on chapter V.- Answers to the exercises.

LINEAR equations play an important part, not only in mathe­ matics itself, but also in many fields in which mathematics is used. Whether we deal with elastic deformations or electrical networks, the flutter of aeroplane wings or the estimation of errors by the method of least squares, at some stage in the cal­ culation we encounter a system of linear equations. In each case the problem of solving the equations is the same, and it is with the mathematical treatment of this question that this book is concerned. By meeting the problem in its pure state the reader will gain an insight which it is hoped will help him when he comes to apply it to his field of work. The actual pro­ cess of setting up the equations and of interpreting the solution is one which more properly belongs to that field, and in any case is a problem of a different nature altogether. So we need not concern ourselves with it here and are able to concentrate on the mathematical aspect of the situation. The most important tools for handling linear equations are vectors and matrices, and their basic properties are developed in separate chapters. The method by which the nature of the solution is described is one which leads immediately to a solu­ tion in practical cases, and it is a method frequently adopted when solving problems by mechanical or electronic computers.