MATH0005 Algebra 1 Matthew Towers Contents About these notes 1 Logic 1.1 Propositional calculus 1.2 Well-formed formulas 1.3 Truth tables 1.4 Truth values for WFFs 1.5 Logical equivalence 1.6 Useful logical equivalences 1.7 The contrapositive 1.8 Adequacy 1.9 First order logic Further reading 2 Sets and functions 2.1 Introduction to set theory 2.2 Set operators 2.3 Set algebra 2.4 De Morgan’s laws 2.5 Functions 2.6 Function composition 2.7 Function properties 2.8 Invertibility 2.9 Conditions for invertibility 2.10 Permutations 2.11 Inverses and composition 2.12 Cycles 2.13 Products of disjoint cycles 2.14 Powers and orders 2.15 Transpositions 2.16 Sign Further reading 3 Matrices 3.1 Matrix definitions 3.2 Matrix multiplication 3.3 Transpose 3.4 Multiplication properties 3.5 Invertible matrices 3.6 Systems of linear equations 3.7 Row operations 3.8 Elementary matrices 3.9 Row reduced echelon form 3.10 RREF existence and uniqueness 3.11 Solving RREF systems 3.12 Invertibility and RREF 3.13 Finding inverses Further reading 4 Linear algebra 4.1 Fields 4.2 Vector spaces 4.3 Using the vector space axioms 4.4 Subspaces 4.5 Sums and intersections 4.6 Linear independence 4.7 Spanning sequences 4.8 Bases 4.9 Dimension 4.10 Basis and dimension examples 4.11 Extending to a basis 4.12 Finding dimensions 4.13 Linear maps 4.14 Kernel and image 4.15 The rank-nullity theorem 4.16 Matrix nullspace basis 4.17 Column space basis 4.18 Matrix of a linear map 4.19 Matrix of a composition 4.20 Change of basis 4.21 Coordinate isomorphisms Further reading