MA1006: ALGEBRA (2018-2019)

The basic course includes a discussion of the following topics: complex numbers and the theory of polynomial equations, vector algebra in two and three dimensions, systems of linear equations and their solution, matrices and determinants.

Syllabus

• Solving equations.
• Polynomial equations and their roots, polynomial long division, the Rational root theorem.
• Introduction to complex numbers. The addition, subtraction, multiplication and division of
• Complex numbers. Modulus and Argument and the representation of such numbers on an Argand diagram. Loci and regions in the Argand diagram. De Moivre’s theorem and applications. Complex exponential, logarithm, sine and cosine.
• Systems of linear equations, Gaussian elimination.
• Matrix algebra. Determinants of square matrices (of any dimension). Matrix inversion (the cofactor method and Gaussian elimination).
• Vectors and linear maps. Special matrices (e.g rotation matrices). Matrix design. Eigenvalues and eigenvectors.
• Topics from: Diagonalizability. Subspaces, dimensions and linear independence. The rank and nullity of a matrix.

Course Aims

The overall aim of the course is to introduce the student to some important topics in basic algebra

and mathematics. These topics are polynomial equations, complex numbers, vectors and matrices. We aim:

• To introduce basic logical concepts and proper notation used in mathematics.

• To study certain standard results in polynomial equations and to use these to motivate the need for the extension of the "real" number system to the system of complex numbers.

• To study the algebra and geometrical representation of complex numbers and to introduce De Moivre’s Theorem.

• To introduce matrices and the algebra thereof, including determinants and matrix inverses, and the application of matrices to solving linear equations.

• To study the geometry of matrices and the linear transformations they induce, including the concepts of eigenvalue and eigenvector.

Learning Objectives

By the end of the course the student should:

• Have an understanding of the need of precision in mathematics and have a working knowledge of basic logical rules.

• Have a clear understanding of the necessity of complex numbers and be familiar with their elementary manipulation and geometrical representation.

• Be able to carry out division of polynomials and solve polynomial equations up to degree three.

• Be familiar with the concept of a matrix, and able to solve systems of linear equations using Gaußian elimination.

• Be able to perform calculations involving matrices, such as matrix inversion, finding eigenvalues and eigenvectors, and diagonalization.

Information on contact teaching time is available from the course guide.

More Information about Week Numbers

1st Attempt: 1 two-hour written examination (70%) and in-course assessment (30%).

Resit: 1 two-hour written examination paper (maximum of (100%) resit and (70%) resit with (30%) in-course assessment).

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination.

Support tutorials to be arranged by the Course Coordinator, as need arises.