Vector Spaces, Subspaces, Linear Independence, Span, Basis, Dimension, Linear Transformations, Orthogonal Transformations, Inner product, Orthogonal projections, Vector and matrix norms, Matrix subspaces, Diagonalization, Eigen decomposition, SVD, Least Squares, Pseudo inverse. Linear system of equations, Solution, Numerical methods of solution. Iterative refinement, QR factorization, Gram Schmidt Orthogonalization, Cholesky decomposition. conditioning and condition number, stability of numerical algorithms, vector and matrix norms, convergent matrices, stability of nonlinear systems, sensitivity analysis. QR method, Power method and applications, Jacobi method for finding the eigenvalues of a given matrix. Cholesky QR algorithm Quadratic forms. Positive definite and positive semidefinite matrices. computational linear algebra, including solution of a system of linear equations, least-squares solutions of linear systems, computation of eigenvalues, eigenvectors, and singular value problems. Applications from numerous disciplines of science and engineering,