Lecture 8 (Part 2): The relation between the geometric and algebraic multiplicities of an eigenvalue

These are the lectures on Advanced Linear Algebra, taught to BS-IV Mathematics students, which are recorded in order to facilitate students who are at home due to COViD-19. The lectures are started from the last topic that we covered physically in class. Linear algebra is a fundamental tool in many fields, including mathematics and statistics, computer science, economics, and the physical and biological sciences. This course offers a complete second course in linear algebra, tailored to help students transition from basic theory to advanced topics and applications. Topics include Theory of Inner Product Spaces; Orthonormal Systems; The Gram–Schmidt Process; The Riesz Representation Theorem; Adjoints of Linear Transformations and Matrices; Parseval’s Identity and Bessel’s Inequality; Isometries on an Inner Product Space; Unitary Matrices; Permutation Matrices; Orthogonal Complements; Orthogonal Projections; Best Approximation; Eigenvalues and eigenvectors; Minimal Polynomial; Multiplicities of eigenvalues; Diagonalizing Linear Transformations and applications; The Jordan Normal Form; Positive Semidefinite Matrices; The Square Root of a Positive Semidefinite Matrix; Simultaneous Diagonalization of Quadratic Forms; Introduction to multilinear maps; tensors; tensor product spaces; exterior spaces and differential forms. @RUeamHK0X6#