Matrix Calculations: Basis Change and Extensions

This lecture covers the calculation of matrices in the canonical basis, basis change formulas, and the concept of extensions in finite-dimensional vector spaces. It also explores the properties of field extensions, including the degree of extensions and quadratic extensions. The lecture delves into the base change formula, the modulus of complex numbers, and the concept of complex conjugation. Additionally, it discusses the polar decomposition of complex numbers and the isomorphism of polar coordinates. The lecture concludes with the definitions of cosine, sine, and argument in complex numbers.