Selected Topics in Algebra

The module begins with a brief explanation of how to introduce rational numbers based on integers. Subsequently, complex numbers are introduced as pairs of real numbers and basic operations on these pairs are defined including addition, subtraction, The symbol i (the square root of -1) and the algebraic form of complex numbers are defined. Later we discuss quadratic equations using the quadratic formula, which is still valid among the complex numbers. We state the theorem that every non-constant complex polynomial of degree n has exactly n roots, counted with multiplicity and using complex conjugation we prove that a real polynomial of odd degree always has a real root. We introduce the trigonometric form of complex numbers. We discuss the geometric meaning of addition, multiplication, conjugation. The complex exponential function and its relation to trigonometric functions and the new way of representing a complex number are discussed as well as domain colouring of some complex functions. Expressing linear transformations using complex numbers is introduced. In the final sessions, we will talk about matrices expressing a general (linear) transformation and the addition of matrices. We will discuss the meaning of the determinant as area/volume. We will investigate the existence of a unique solution to a system of linear equations depending on the determinant of the matrix. Matrices of transformations will be looked at in different bases, taking projection as an example. Eigenvectors and eigenvalues are introduced as a simple basis. We will also cover techniques of finding eigenvalues and eigenvectors in simple cases and the explicit formula for the Fibonacci sequence using matrices.