4 Credits (3-1-0)

Vector spaces over Q, R and C, subspaces, linear independence, linear span of a set of vectors, basis and dimension of a vector space, sum and direct sum. Systems of linear (homogeneous and non-homogeneous) equations, matrices and Gauss elimination, elementary row operations, row space, column space, null space and rank of a matrix. Linear transformation, rank-nullity theorem and its applications, matrix representation of a linear transformation, change of basis and similarity. Eigenvalues and eigenvectors, characteristic and minimal polynomials, Cayley-Hamilton theorem (without proof) and applications. Review of first order differential equations, Picardâ€™s theorem, linear dependence and Wronskian. Dimensionality of space of solutions, linear ODE with constant coefficients of second and higher order, Cauchy-Euler equations, Method of undetermined coefficients and method of variation of parameters. Boundary Value Problems: SturmLiouville eigenvalue problems. System of linear differential equations with constant coefficients, fundamental matrix, matrix methods. Power Series and its convergence, power series method, Fourier series, Laplace Transform Method.