MATH 105 - Linear Algebra Dr Tony Nixon Contents: 1 Matrices 1.1 Motivation and definitions 1.2 Matrix addition and scalar multiplication 1.3 Vector multiplication 1.4 Matrix multiplication 1.5 The transpose of a matrix 1.6 Exercises 2 Row operations on matrices 2.1 Elementary matrices 2.2 Row reduction of matrices to echelon form 2.3 Rank of a matrix 2.4 Exercises 3 Inverting matrices 3.1 Identity matrices and inverses 3.2 Inverting matrices 3.3 Invertible matrices as a product of elementary matrices 3.4 Exercises 4 Determinants 4.1 Determinants of 2×2 matrices 4.2 Determinants of square matrices 4.3 Properties of determinants 4.4 More on elementary matrices 4.5 Exercises 5 Systems of linear equations 5.1 Introduction 5.2 Row operations 5.3 Number of solutions 5.4 Systems with parameters 5.5 Exercises 6 Linear transformations 6.1 Definition and first examples 6.2 Linear transformations versus matrices 6.3 Invertible linear transformations 6.4 Transformations of the Euclidean space, ℝ3 6.5 Exercises 7 Eigenvalues and eigenvectors 7.1 Definition of eigenvalue and eigenvector 7.2 How to find the eigenvalues and eigenvectors 7.3 Eigenvalues of a 2×2 matrix 7.4 Subspaces and dimension 7.5 Exercises