Linear Algebra II

General vector spaces and linear matrices. Sum and direct sum of vector spaces. Dimension formula. Nilpotent and cyclic matrices. Tensor products of vector spaces and linear matrices. Diagonalisation of linear operators. Inner product spaces. Introduction to Hilbert spaces. Isometries. Riesz representation theorem and adjoint operators. Self-adjoint operators. Spectral theorem. Multilinear and quadratic forms. The norm of a linear operator. Positive definite and positive semidefinite operators. Sylvester's law of inertia. Generalized eigenvectors. Jordan normal form of matrix. Square roots of matrices. Minimal polynomial and Hamilton-Cayley's theorem. Matrix exponential function. Systems of linear differential equations with constant coefficients. Students carry out a minor project individually.