Unit-1 Definition and Examples of Vector Spaces, Subspaces, Sum and direct sum of subspaces, Linear span, Linear dependence, Independence and their basic properties. Basis, Existence theorem for basis, Existence theorem, Invariance of the number of elements of a basis. Dimension, Finite dimensional vector spaces, Existence of complementary subspaces of a subspace of a finite dimensional vector space, Dimension of sum of subspaces, Quotient space and its dimension.

Unit-2 Linear transformations and their representation as matrices, Algebra of linear transformations, Rank-Nullity theorem, Change of basis, dual space, Bi-dual space and natural isomorphism, Adjoint of a linear transformation. Eigen values and Eigen vectors of a linear transformation, Diagonalisation, Bilinear, Quadratic and Hermitian forms.

Unit-3 Inner Product Space : Cauchy-Schwar'z inequality, Orthogonal vectors, Orthogonal complements, Orthonormal sets and bases, Bessel's inequality for finite dimensional spaces. Gram-Schmidt orthogonalization process.

Unit-4 Solution of Equations : Bisection, Secant, Regula-Falsi, Newton's methods. Roots of second degree polynomials.

Interpolation : Lagrange interpolation, Divided differences, Interpolation formula using differences, Numerical quadrature, Newton-Cote's formulae, Gauss quadrature formulae.

Unit-5 Linear Equations : Direct methods for solving systems of linear equations (Gauss elimination, LU decomposition, Cholesky decomposition), Iterative methods (Jacobi, Gauss-Seidel reduction methods).

Ordinary Differential Equations : Euler method, Single-step method, Runge-Kutta's method, Multi-step methods. Milne-Simpson method, Methods based on Numerical integration, Methods based on Numerical differentiation.

GAJENDRA UJJAINKAR