1 Fields and complex numbers

1.1 Reminders on sets and mappings

Exercises

1.2 Fields

1.3 Complex numbers

Exercises

2 Matrices, I

2.1 Definitions

2.2 Arithmetic with matrices

2.3 Transpose and inverse

Exercises

3 Matrices, II

3.1 Row echelon form

3.2 Solving equations

3.3 Inverting a matrix

Exercises

4 Vector spaces

4.1 Abstract vector spaces

4.2 Linear combinations

4.3 Vector subspaces

4.4 Subspaces generated by sets

4.5 Generating sets and finite-dimensionality

4.6 Linear independence

Exercises

5 Dimensions of vector spaces

5.1 Growing and shrinking sets

5.2 The fundamental inequality

5.3 Bases of vector spaces

5.4 Dimensions of vector spaces

5.5 Computing with subspaces

Exercises

6 Linear maps

6.1 Linear maps

6.2 Images, preimages, kernels

6.3 The rank-nullity theorem

Exercises

7 Linear maps and matrices

7.1 Linear mappings associated to matrices

7.2 Computing kernels and images

Exercises

8 Coordinate systems and changes of basis

8.1 Linear coordinate systems

8.2 The matrix of a linear map

8.3 Change of basis

Exercises

9 The determinant, I

9.1 Axiomatic characterisation

9.2 Uniqueness of the determinant

9.3 Existence of the determinant

Exercises

10 The determinant, II

10.1 Properties of the determinant

10.2 Permutations

10.3 The Leibniz formula

10.4 Cramer’s rule

Exercises

11 Endomorphisms, I

11.1 Matrices of endomorphisms

11.2 Detour: More on Subspaces

11.3 Eigenvectors and eigenvalues

11.4 The characteristic polynomial

Exercises

12 Endomorphisms, II

12.1 Properties of eigenvalues

12.2 Special endomorphisms

Exercises

13 Affine spaces and quotient vector spaces

13.1 Affine mappings and affine spaces

13.2 Quotient vector spaces

Exercises