Book:P.M. Cohn/Linear Equations

Subject Matter

 * Linear Algebra

Contents

 * Preface


 * Introduction


 * 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 $I$
 * 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 $V$


 * Answers to the exercises


 * Index