# Book:P.M. Cohn/Linear Equations

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## P.M. Cohn:

## P.M. Cohn: *Linear Equations*

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

### Subject Matter

### Contents

- Preface

- Introduction

- chapter
- 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 by a finite number of vectors**Exercises on chapter $I$*

- 1.

- 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 $II$*

- 1--2.

- 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 $III$*

- 1--2.

- 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 $IV$*

- 1--2.

- 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 $V$*

- 1.

- Answers to the exercises

- Index

## Cited by

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space: Example $62$: Footnote