MA3202 Linear Algebra and Numerical Methods Syllabus:
MA3202 Linear Algebra and Numerical Methods Syllabus – Anna University Regulation 2021
OBJECTIVES:
The basic concepts and tools of the subject covered are:
• Vector spaces and subspaces; linear independence and span of a set of vectors, basis and dimension; the standard bases for common vector spaces;
• Linear maps between vector spaces, their matrix representations, null-space and Range spaces, the Rank- Nullity Theorem;
• Inner product spaces: Cauchy-Schwarz inequality, orthonormal bases, the GrammSchmidt procedure, orthogonal complement of a subspace, orthogonal projection;
• Eigenvalues and eigenvectors, diagonalizability of a real symmetric matrix, canonical forms;
• Mathematical foundations of numerical techniques for solving linear systems, eigenvalue problems and generalized inverses.
UNIT I VECTOR SPACES
Vector spaces – Subspaces – Linear combinations – Linear Span – Linear dependence – Linear independence – Bases and Dimensions
UNIT II LINEAR TRANSFORMATIONS
Linear Transformation – Null space, Range space – dimension theorem – Matrix and representation of Linear Transformation – Eigenvalues Eigenvectors of linear transformation – Diagonalization of linear transformation – Application of diagonalization in linear system of differential equations.
UNIT III INNER PRODUCT SPACES
Inner Products and norms – Inner Product Spaces – Orthogonal vectors – Gram Schmidt orthogonalization process – Orthogonal complement – Least square Approximations
UNIT IV NUMERICAL SOLUTION OF LINEAR SYSTEM OF EQUATIONS
Solution of linear system of equations – Direct methods: Gauss elimination method – Pivoting, Gauss Jordan method, LU decomposition method and Cholesky decomposition method – Iterative methods: Gauss-Jacobi Method, Gauss-Seidel Method and SOR Method
UNIT V NUMERICAL SOLUTION OF EIGENVALUE PROBLEMS AND GENERALISED INVERSES
Eigen value Problems: Power method – Jacobi‘s rotation method – Conjugate gradient method – QR decomposition – Singular value decomposition method.
TOTAL:60 PERIODS
OUTCOMES:
▪ The students will be able to solve system of linear equations, to use matrix operations and vector spaces using algebraic methods.
▪Demonstrate understanding of common numerical methods and how they are used to obtain approximate solutions.
• Apply numerical methods to obtain approximate solutions to mathematical problems.
• Derive numerical methods for various mathematical operations and tasks, such as interpolation, differentiation, integration, the solution of linear and nonlinear equations, and the solution of differential equations.
• Analyze and evaluate the accuracy of common numerical methods.
TEXT BOOKS:
1. Faires, J.D. and Burden, R., “Numerical Methods”, Brooks/Cole (Thomson Publications), 4th Edition, New Delhi, 2012.
2. Friedberg, S.H., Insel, A.J. and Spence, E., “Linear Algebra”, Pearson Education, 5th Edition, New Delhi, 2008.
3. Williams, G, “Linear Algebra with Applications”, Jones & Bartlett Learning, First Indian Edition, New Delhi, 2019.
REFERENCES:
1. Bernard Kolman, David R. Hill, “Introductory Linear Algebra”, Pearson Education, First Reprint, New Delhi, 2010.
2. Gerald, C.F, and Wheatley, P.O., “Applied Numerical Analysis”, Pearson Education, 7th Edition, New Delhi, 2004.
3. Kumaresan, S., “Linear Algebra – A geometric approach”, Prentice – Hall of India, Reprint, New Delhi, 2010.
4. Richard Branson, “Matrix Operations”, Schaum’s outline series, McGraw Hill, New York, 1989.
5. Strang, G., “Linear Algebra and its Applications”, Cengage Learning, New Delhi, 2005.
6. Sundarapandian. V, “Numerical Linear Algebra”, Prentice – Hall of India, New Delhi, 2008.