Linear Equations

1. Vectors. - 1. Notation. - 2. Definition of vectors. - 3. Addition of vectors. - 4. Multiplication by a scalar. - 5. Geometrical interpretation. - 67. 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. - 12. Regular systems. Notations and statement of results. - 34. Elementary operations on systems. - 57. Proof of the Main Theorem. - 89. 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. - 12. 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. - 12. 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. Language: English