Linear Algebra (Math 110), Summer 2012

Basic Set Theory......Page 2 Functions......Page 5 Fields......Page 8 Basic Definitions......Page 9 Subspaces......Page 16 Linear Dependence & spanK......Page 20 Linear Morphisms, Part I......Page 25 Bases, Dimension......Page 30 Finding a basis......Page 35 Solving problems......Page 39 Change of basis/change of coordinates......Page 40 Linear morphisms II......Page 42 Rank, classification of linear morphisms......Page 47 Dual Spaces (non-examinable)......Page 50 Coordinate-free systems of equations or Why row-reduction works......Page 53 Eigenthings......Page 55 Characteristic polynomial, diagonalising matrices......Page 57 Invariant subspaces......Page 61 Nilpotent endomorphisms......Page 63 Determining partitions associated to nilpotent endomorphisms......Page 67 Algebra of polynomials......Page 69 Canonical form of an endomorphism......Page 74 The Jordan canonical form......Page 80 Bilinear forms......Page 84 Nondegenerate bilinear forms......Page 88 Adjoints......Page 91 Real and complex symmetric bilinear forms......Page 93 Computing the canonical form of a real nondegenerate symmetric bilinear form......Page 96 Euclidean spaces......Page 99 Orthogonal complements, bases and the Gram-Schmidt process......Page 106 Hermitian spaces......Page 110 Normal morphisms......Page 114 Self-adjoint operators and the spectral theorem......Page 116