Lectures in course MATH-115(b): Advanced linear algebra II

Explores causal relations, inertial universes, and proper time in spacetime.

Explores the definition and properties of Hermitian forms in complex vector spaces.

Explores Hermitian operators, auto-adjoint properties, and spectral theorems in Hermitian spaces.

Explains the postulates of quantum mechanics and the representation of observables by operators.

Explores matrix recurrence relations, eigenvalues, Fibonacci sequence, and the golden ratio in linear algebra.

Explores the properties and applications of triangular matrices in linear algebra.

Explores quadratic forms, symmetric bilinear forms, and their properties.

Covers the definition of scalar product, properties, examples, and applications in Euclidean spaces, including the Cauchy-Schwartz inequality.

Covers the concept of orthogonality between vectors and the notion of distance between functions.

Explores orthogonality, scalar product, and orthonormal bases in vector spaces.

Covers the fundamentals of polynomials, endomorphisms, division, roots, matrices, and algebraic homomorphisms.

Explores the uniqueness of minimal polynomials and the division algorithm for polynomials.

Explores division polynomials, theorems, spectral values, and minimal polynomials in endomorphisms and vector spaces.

Explains orthogonal projection and vector decomposition with examples in particle trajectory analysis.

Covers the theory and applications of orthogonal projections, focusing on the Gram-Schmidt process for orthonormalization.

Explores isometries in Euclidean spaces, including translations, rotations, and linear symmetries, with a focus on matrices.

Covers the Jordan normal form and the decomposition of vector spaces by an endomorphism.

Covers Jordan form, diagonalization, and nilpotent matrices, discussing properties and decomposition.

Covers the theory and examples of Jordan forms and their practical application in matrix analysis.

Explores eigenvalues and minimal polynomial, emphasizing their importance in linear algebra.