cover -- copyright page -- title page -- Preface -- Notational Conventions -- Contents -- 1 Symmetries of vector spaces -- 1.1 What is a symmetry? -- 1.2 Distance is fundamental -- 1.3 Groups of symmetries -- 1.4 Bilinear forms and symmetries of spacetime -- 1.5 Putting the pieces together -- 1.6 A broader view: Lie groups -- 2 Complex numbers, quaternions and geometry -- 2.1 Complex numbers -- 2.2 Quaternions -- 2.3 The geometry of rotations of R^3 -- 2.4 Putting the pieces together -- 2.5 A broader view: octonions -- 3 Linearization -- 3.1 Tangent spaces -- 3.2 Group homomorphisms -- 3.3 Differentials -- 3.4 Putting the pieces together -- 3.5 A broader view: Hilbert's fifth problem -- 4 One-parameter subgroups and the exponential map -- 4.1 One-parameter subgroups -- 4.2 The exponential map in dimension 1 -- 4.3 Calculating the matrix exponential -- 4.4 Properties of the matrix exponential -- 4.5 Using exp to determine L.G/ -- 4.6 Differential equations -- 4.8 A broader view: Lie and differential equations -- 4.9 Appendix on convergence -- 5 Lie algebras -- 5.1 Lie algebras -- 5.2 Adjoint maps-big 'A' and small 'a' -- 5.3 Putting the pieces together -- 5.4 A broader view: Lie theory -- 6 Matrix groups over other fields -- 6.1 What is a field? -- 6.2 The unitary group -- 6.3 Matrix groups over finite fields -- 6.4 Putting the pieces together -- Suggestions for further reading -- 6.5 A broader view: finite groups of Lie type and simple groups -- Appendix I Linear algebra facts -- Appendix II Paper assignment used at Mount Holyoke College -- Appendix III Opportunities for further study -- Metric vector spaces and symmetries -- Lie algebras and Chevalley groups -- Quaternions and octonions -- Connections to physics -- Lie groups as manifolds -- Solutions to selected problems -- 1. Symmetries of vector spaces -- 1.1. What is a symmetry?.

1.2. Distance is fundamental -- 1.3. Groups of symmetries -- 1.4. Bilinear forms and symmetries of spacetime -- 2. Complex numbers, quaternions and geometry -- 2.1. Complex numbers -- 2.2. Quaternions -- 2.3. The geometry of rotations of R^3 -- 3. Linearization -- 3.1. Tangent spaces -- 3.2. Group homomorphisms -- 3.3. Differentials -- 4. One-parameter subgroups and the exponential map -- 4.1. One-parameter subgroups -- 4.2. The exponential map in dimension 1 -- 4.3. Calculating the matrix exponential -- 4.4. Properties of the matrix exponential -- 4.5. Using exp to determine L.G/ -- 4.6. Differential equations -- 5. Lie algebras -- 5.1. Lie algebras -- 5.2. Adjoint maps-big 'A' and small 'a' -- 6. Matrix groups over other fields -- 6.1. What is a field? -- 6.2. The unitary group -- 6.3. Matrix groups over finite fields -- Bibliography -- Index -- Notation -- About the Author.