# Book:P.M. Cohn/Linear Equations

## P.M. Cohn: Linear Equations

Published $\text {1958}$, Routledge & Kegan Paul.

### Contents

Preface
Introduction
chapter
1. Vectors
1. Notation
2. Definition 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 by a finite number of vectors
Exercises on chapter $I$
2. The Solution of a System of Equations: the Regular Case
1--2. Regular systems. Notations and statements 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 by 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 $III$
4. 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 a homogeneous system
8. Illustrations
9. The solution of general systems
10. Illustrations
11. Geometrical interpretation
Exercises on chapter $IV$
5. Determinants
1. Motivation
2. The $\mathcal 2$-dimensional case
3. The $\mathcal 3$-dimensional case
4. The rule of signs in the $\mathcal 3$-dimensional case
5. Permutations
6. The Kronecker $\varepsilon$-symbol
7. The determinant of an $n \times n$ matrix
8. Cofactors 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 $V$