# MA8352 Linear Algebra and Partial Differential Equations Previous Year Question Paper

### OBJECTIVES:

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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