Lectures related to Pseudo-Euclidean Spaces: Isometries and Bases

Covers the properties and operations of vector spaces, including addition and scalar multiplication.

Explores symmetric matrices, quadratic forms, and critical points in functions of two variables.

Explores the classification of quadratic forms based on eigenvalues and orthogonal diagonalization of symmetric matrices.

Explains linear independence, bases, and dimension in vector spaces, including the importance of the order of vectors in a basis.

Covers tensor products, symmetric power, and exterior power of vector spaces, including properties and applications.

Covers determining vector spaces, calculating kernels and images, defining bases, and discussing subspaces and vector spaces.

Explores bases, dimensions, and matrix ranks in vector spaces with practical examples and proofs.

Explores integral representations, quadratic lattices, and the Hasse Principle.