MATH220 Linear Algebra Mark MacDonald Michaelmas 2017-18 Lancaster University Contents: 1 Fields and Matrices 1.A Fields 1.B Row operations and reduced echelon form 1.C Inverse matrices and triangular matrices Exercises 2 Vector spaces 2.A Vector spaces 2.B Subspaces and spanning sequences 2.C Linear independence 2.D Dimension and bases 2.E Coordinates 2.F Row space and column space Exercises 3 Inner products 3.A Bilinear forms 3.B Positive definiteness 3.C The Cauchy-Schwarz inequality 3.D Orthogonality 3.E The Gram-Schmidt process Exercises 4 Linear transformations 4.A The matrix of a linear transformation 4.B Eigenvalues and eigenvectors 4.C Images and kernels 4.D Dimension theorem 4.E Systems of linear equations 4.F Injective, surjective, and bijective transformations 4.G Change of basis matrices 4.H Diagonalizable matrices Exercises 5 Spectral decomposition 5.A Orthogonal matrices 5.B Real symmetric matrices 5.C Matrix square roots Exercises 6 Jordan normal form 6.A The Cayley-Hamilton theorem 6.B Minimal polynomials 6.C Generalized eigenspaces 6.D Jordan chains and Jordan bases 6.E Jordan normal form 6.F Jordan normal form and the minimal polynomial Exercises 7 Coursework 7.A Week 1 7.B Week 2