· To introduce the basic notions of groups, rings, fields which will then be used to solve related problems.
· To understand the concepts of vector space, linear transformations and diagonalization.
· To apply the concept of inner product spaces in orthogonalization.
· To understand the procedure to solve partial differential equations.
· To give an integrated approach to number theory and abstract algebra, and provide a firm basis for further reading and study in the subject.
UNIT I VECTOR SPACES 12
Vector spaces – Subspaces – Linear combinations and linear system of equations – Linear independence and linear dependence – Bases and dimensions.
UNIT II LINEAR TRANSFORMATION AND DIAGONALIZATION 12
Linear transformation – Null spaces and ranges – Dimension theorem – Matrix representation of a linear transformations – Eigenvalues and eigenvectors – Diagonalizability.
UNIT III INNER PRODUCT SPACES 12
Inner product, norms – Gram Schmidt orthogonalization process – Adjoint of linear operations – Least square approximation.
UNIT IV PARTIAL DIFFERENTIAL EQUATIONS 12
Formation – Solutions of first order equations – Standard types and equations reducible to standard types – Singular solutions – Lagrange‘s linear equation – Integral surface passing through a given curve – Classification of partial differential equations – Solution of linear equations of higher order with constant coefficients – Linear non-homogeneous partial differential equations.
UNIT V FOURIER SERIES SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 12
Dirichlet‘s conditions – General Fourier series – Half range sine and cosine series – Method of separation of variables – Solutions of one dimensional wave equation and one-dimensional heat equation – Steady state solution of two-dimensional heat equation – Fourier series solutions in Cartesian coordinates.
TOTAL: 60 PERIODS
Upon successful completion of the course, students should be able to:
· Explain the fundamental concepts of advanced algebra and their role in modern
mathematics and applied contexts.
· Demonstrate accurate and efficient use of advanced algebraic techniques.
· Demonstrate their mastery by solving non – trivial problems related to the concepts and by
proving simple theorems about the statements proven by the text.
· Able to solve various types of partial differential equations.
Able to solve engineering problems using Fourier series.
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2. James, G. ―Advanced Modern Engineering Mathematics‖, Pearson Education, 2007.
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5. Lay, D.C., ―Linear Algebra and its Applications‖, 5th Edition, Pearson Education, 2015.
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