Abstract
This chapter is devoted mainly to Euclidean vector spaces and their transformations. It starts with notions of inner product, length, angle, Gramian, orthogonality, orthonormal basis, etc. Orthogonal transformations are investigated, and orientation of Euclidean spaces is discussed. After that, symmetric linear transformations of real vector spaces are investigated in greater detail; for instance, a basic property of such transformations to have a real eigenvalue and real eigenvector is provided with three proofs, based on different principles, and the extremal properties of the eigenvalues of a symmetric linear transformation are discussed (the Courant–Fischer theorem). Theoretical results and notions are accompanied with numerous examples from different areas, including mechanics and geometry; examples include elements of the theory of hypersurfaces in a Euclidean space (principal curvatures, Euler’s formula, etc.). At the end of the chapter, pseudo-Euclidean vector spaces and Lorentz transformations (analogue of orthogonal transformations for pseudo-Euclidean spaces) are considered in greater detail.