1 Preliminaries
1.1 Sets
1.2 Equivalence Relations
1.3 Functions
1.4 Practice Problems
2 Matrices
2.1 Definition and Notation
2.2 Types of Matrices
2.3 Matrix Algebra
2.4 Non-Singular Matrices
2.5 Elementary Row Operations on Matrices
2.6 Elementary Matrices
2.7 Normal Form of a Matrix
2.8 Practice Problems
3 Systems of Linear Equations
3.1 Substitution and Gaussian Elimination
3.2 Elementary Row Operations and Systems of Linear Equations
3.3 Homogeneous Systems of Linear Equations
3.4 Practice Problems
4 Determinants
4.1 2 by 2 Determinants
4.2 3 by 3 Determinants
4.3 \(n\) by \(n\) Determinants
4.4 Practice Problems
5 Vector Spaces
5.1 Definition and Examples of Vector Spaces
5.2 Vector Subspaces
5.3 Linear Combinations
5.4 Basis and Dimension
5.5 Coordinates and the Transition Matrix
5.6 Practice Problems
6 Linear Transformations
6.1 Definition and Examples
6.2 Null Spaces and Ranges
6.3 The Matrix of a Linear Transformation
6.4 Invertibility
6.5 Practice Problems
7 Orthogonality
7.1 Inner Product Spaces
7.2 Orthogonal Vectors
7.3 Practice Problems
8 Characteristic Roots and Vectors
8.1 Eigenvalues and Vectors
8.2 Diagonalisation of Matrices and Linear Transformations
8.3 Cayley-Hamilton Theorem
8.4 Diagonalisation of Symmetric Matrices
8.5 Practice Problems
9 Quadratic Forms
9.1 Introduction
9.2 Diagonal Forms
9.3 Conic Sections
9.4 Practice Problems
Further reading