MA8352 Linear Algebra and Partial Differential Equations Previous Year Question Paper
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
Vector spaces – Subspaces – Linear combinations and linear system of equations – Linear independence and linear dependence – Bases and dimensions.
UNIT II LINEAR TRANSFORMATION AND DIAGONALIZATION
Linear transformation – Null spaces and ranges – Dimension theorem – Matrix representation of a linear transformations – Eigenvalues and eigenvectors – Diagonalizability.
UNIT III INNER PRODUCT SPACES
Inner product, norms – Gram Schmidt orthogonalization process – Adjoint of linear operations – Least square approximation.
UNIT IV PARTIAL DIFFERENTIAL EQUATIONS
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
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.
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