Applied Linear Algebra

Vector spaces, linear dependence, basis. Representation of linear transformations with respect to a basis. Inner product spaces, Hilbert spaces, linear functions, the Riesz representation theorem and adjoints. Orthogonal projections, products of projections, orthogonal direct sums. Unitary and orthogonal transformations, complete orthonormal sets and Parseval"s identity. Closed subspaces and the projection theorem for Hilbert spaces. Polynomials. The algebra of polynomials, matrix polynomials, annihilating polynomials and invariant subspaces, Jordan forms. Applications : Complementary orthogonal spaces in networks, properties of graphs and their relation to vector space properties of their matrix representations. Solution of state equations in linear system theory. Relation between the rational and Jordan forms. Numerical linear algebra : Direct and iterative methods of solutions of linear equations. Matrices, norms, complete metric spaces and complete normal linear spaces (Banach spaces). Least squares problems (constrained and unconstrained). Eigenvalue problem.