Quadratic Forms and Matrices

PrefaceTranslator's NoteChapter I General Theory of Quadratic Curves 1. Transformation of Coordinates in the Plane 2. Reduction to Canonical Form of the Equation of a Quadratic Curve with Center at the Origin 3. Invariants and Classification of Quadratic Forms in Two Variables 4. Reduction to Canonical Form of the General Equation of a Quadratic Curve 5. Equation of the Center. Test for Degeneracy of a Quadratic Curve. ExamplesChapter II General Theory of Quadric Surfaces 6. Transformation of Rectangular Coordinates in Space 7. Some General Deductions Based on the Formulas for the Transformation of Coordinates 8. Reduction to Canonical Form of the Equation of a Quadric with Center at the Origin 9. Invariants and Classification of Quadratic Forms in Three Variables 10. Reduction to Canonical Form of the General Equations of a Quadric Surface 11. Equation of the Center. Test for Degeneracy of a Quadric. ExamplesChapter III Linear Transformations and Matrices 12. Linear Transformations of the Plane 13. Multiplication of Linear Transformations of the Plane and of Square Matrices of Order Two. Addition of Matrices. Multiplication of a Matrix by a Number 14. A Theorem of the Determinant of a Product of Two Matrices 15. Geometric Meaning of the Determinant of a Linear Transformation. Singular Transformations 16. The Inverse of a Linear Transformation of the Plane 17. Change of the Coordinates of a Vector Due to a Change of Basis 18. Change of the Matrix of a Linear Transformation Due to a Change of Basis 19. Matrix Form of a System of Two Linear Equations 20. Linear Transformations in Three-Dimensional Space and Square Matrices of Order Three 21. Eigenvectors of a Linear Transformation 22. Characteristic Equation of a Matrix of a Linear Transformation 23. Symmetric Linear Transformations. Reduction to Canonical Form of the Matrix of a Symmetric Linear Transformation 24. Reduction to Canonical Form of the Matrix of a Symmetric Linear Transformation of Space 25. Reduction of a Quadratic Form to Canonical Form. Application to the Theory of Quadratic Curves and Quadric SurfacesAppendix I Vectors. Operations on Vectors and Some of Their PropertiesAppendix II Elements of the Theory of Determinants 1. Determinants of Order Two and Systems of Two Linear Equations in Two Unknowns 2. Homogeneous System of Two Linear Equations in Three Unknowns 3. Determinants of Order Three 4. Cofactors and Minors 5. Systems of Three Linear Equations in Three UnknownsSubject Index