A Syllabus of Linear Algebra. 1. 2. 3. 4.
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Course’s name: Linear algebra Credit : 3 Lecturer : Ph.D. Dang Van Vinh Goals of the course: This is an introductory course emphasizing techniques of linear algebra and its applications: Matrix operations, determinants, linear equations, vector spaces, linear transformations, eigenvalues, and eigenvectors, quadratic form, orthogonality. Student’s tasks. - To attend the whole course. - Be prepared for the next lesson. - Do any assigned homeworks. Literature: 1. V.A. Ilyin and E.G. Poznyak, Linear algebra; Mir Publishers Moscow, 1986. 2. Greub W.H. Linear algebra, 3ed., Springer, 1967. 3. Hefferon J. Linear algebra, 4. Kuttler K. Introduction to linear algebra for mathematicians, 5. Meyer C.D. Matrix analysis and applied linear algebra, SIAM, 2000. 6. Nicholson W.K. Linear algebra with applications , 3ed, PWS Boston, 1993. 7. Strang G. Linear algebra and its applications, 3ed., Thomson, 1988. 8. Kaufman L. Computational Methods of Linear Algebra , 2ed., Wiley,2005. 9. Steven Leon, Linear Algebra with Applications, 7th Edition, Pearson Prentice Hall, 2006 10. David C. Lay, Linear Algebra and its applications, Addison - Wesley Publishing Company, New York, 1993. 11. Proskuriyakov I.V. Problems in Linear algebra. A standard for estimation. - A middle test:(20%). - Assignments: (30%) - A final Examination: (80%). A concrete content of the subject.
Week 1
Topics Chapter 0. A Complex numbers
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Chapter 1. Matrices and Determinants
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Chapter 1. (cont)
Contents The standard (algebraic) form; The trigonometric form; The exponent of complex numbers; The root of complex numbers; The fundamental theorem of algebra. A matrix algebra: a definitions; matrix operations; elementary row operations; a row reduction and echelon forms. The concept of a determinant: the recursive definition; properties of determinants; Methods of calculation; Laplace’s expansion. An invertible matrix: The concept of an inverse matrix, properties; two methods of finding of an inverse of a matrix : formula and elementary row operations;
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Chapter 1 (cont) Chapter 2. Systems of linear equations. Chapter 2 (cont) Chapter 3. Linear Spaces
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Chapter 3 (cont).
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Chapter 3 (cont)
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Chapter 4. Euclidean spaces.
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Chapter 4 (cont) Chapter 5. Linear transformation. Chapter 5 (cont)
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Chapter 6. Eigenvalues; eigenvectors Chapter 6 (cont) Chapter 7. A quadratic form. Chapter 7. (cont)
The range of a matrix; properties; A method of calculation of range. The concept of a system of linear equations and its solutions. The Kronecker – Capelli Theorem. A general linear system; A quadratic system of linear equations with a nonzero determinant of the system matrix (Crame’s system); A homogeneous system. The concept of a linear space: A definition of a linear space; some properties of a arbitrary linear spaces; The concept of a linear dependence of the elements. The basis and the coordinates of a vector with respect to the basic. Subspace of linear spaces: The concept of a subspace and a linear closure; The sum and the intersection of subspace; an expension of a linear space into a direct sum of subspaces; A null space and a column space. The definition of a scalar product and its properties; a definition of euclidean spaces. Orthonormal sets; Orthogonal subspaces. The concept of an orthonormal basis of a finite dimensional euclidean space. The Gram - Schmidt orthogonalization process. A definition of a linear transformation and examples; a kernel and an image of a linear transformation. Matrix representation of linear transformations. Matrix Eigenvalues and eigenvectors; eigenvalues and eigenvectors of linear transformation. a diagonalization of a matrix; a diagonalization of a linear transformation; a diagonalization of symmetric matrices. The concept of a bilinear form; a representation of a bilinear form in a finite- dimensional linear space; a matrix of a bilinear form. A quadratic form; a canonic form; reducing a quadratic form to the sum of squares: The Lagrange’s method and Jacobi’s method. The law of Inertia of quadratic forms; a classification of quadratic forms; Sylvester’s criterion for the definiteness of a quadratic form.