Vectors and Matrices

Introduction Complex numbers Basic properties Complex exponential function Roots of unity Complex logarithm and power De Moivre's theorem Lines and circles in C Vectors Definition and basic properties Scalar product Geometric picture (R2 and R3 only) General algebraic definition Cauchy-Schwarz inequality Vector product Scalar triple product Spanning sets and bases 2D space 3D space Rn space Cn space Vector subspaces Suffix notation Geometry Lines Plane Vector equations Linear maps Examples Rotation in R3 Reflection in R3 Linear Maps Rank and nullity Matrices Examples Matrix Algebra Decomposition of an n x n matrix Matrix inverse Determinants Permutations Properties of determinants Minors and Cofactors Matrices and linear equations Simple example, 2 x 2 Inverse of an n x n matrix Homogeneous and inhomogeneous equations Gaussian elimination Matrix rank Homogeneous problem Ax = 0 Geometrical interpretation Linear mapping view of Ax = 0 General solution of Ax = d Eigenvalues and eigenvectors Preliminaries and definitions Linearly independent eigenvectors Transformation matrices Transformation law for vectors Transformation law for matrix Similar matrices Diagonalizable matrices Canonical (Jordan normal) form Cayley-Hamilton Theorem Eigenvalues and eigenvectors of a Hermitian matrix Eigenvalues and eigenvectors Gram-Schmidt orthogonalization (non-examinable) Unitary transformation Diagonalization of n x n Hermitian matrices Normal matrices Quadratic forms and conics Quadrics and conics Quadrics Conic sections (n = 2) Focus-directrix property Transformation groups Groups of orthogonal matrices Length preserving matrices Lorentz transformations