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Contents: Determinant formulas

Contents: Existence and uniqueness of the determinant

Contents: Determinant of matrices and operators

Contents: Trace of matrices and operators

Contents: Jordan normal form

Contents: Characteristic and minimal polynomial

Contents: Generalized eigenspaces and multiplicities of eigenvalues

Contents: Generalized eigenvectors and nilpotent operators

Contents: Positive operators and isometries

Contents: Polar decomposition and singular value decomposition

Contents: Spectral theorems

Contents: Self-adjoint and normal operators

Contents: Orthogonal projections

Contents: Orthogonal complement

Contents: Orthonormal bases, Gram-Schmidt procedure

Contents: Inequalities in inner product spaces

Contents: Inner product spaces

Contents: Eigenspaces and diagonalization

Contents: Existence of eigenvalues and upper-triangular matrices

Contents: Invariant subspaces, eigenvalues, eigenvectors

Contents: Rank of a matrix

Contents: Quotient spaces

Contents: Invertibility and basis change

Contents: Invertibility of matrices and elementary matrices