Vectors and Matrices are an essential component of all areas of mathematics and science. This module introduces each of these concepts in preparation for exploring further related concepts in later years of study.  

The basic algebraic properties of Vectors and Matrices are introduced, in conjunction with applications to geometry and the solution of linear systems of equations. All of this material is fundamental to the more abstract setting of Linear Algebra. 

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

• Interpret and manipulate vectors, planes and lines.
• solve sets of linear equations and interpret the solutions
• manipulate matrices and determinants, including the inversion of matrices, both by row operations and by using determinants
• derive and interpret basic ideas of Euclidean vector spaces, including linear independence and basis.                                                                                                

27 - 50 minute lectures

5 - 50 minute tutorial classes

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Ability to work with vectors and matrices, and gain an appreciation of their importance within mathematics.

Transferable Skills:
Ability to understand, manipulate and formulate abstract mathematical concepts.  Critical thinking and problem solving skills.  Ability to present work in a scholarly manner.

Formative assessment is by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures and tutorial classes.

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

Simultaneous linear equations

• Systems of two linear equations in two unknowns. General systems of linear equations in n unknowns. Solution set.
• Reduction of system to echelon form by elementary row operations (Gaussian elimination). Echelon rank. Dimension of solution set.
• Geometric interpretations of systems of equations and solutions sets for the cases of two and three unknowns.

Matrices

• Definitions and basic properties of addition, scalar multiplication and matrix multiplication.
• Transpose of a matrix, inverse matrices and their basic properties.

Matrices and simultaneous linear equations

• Matrix form of a system of linear equations.  Finding solutions with the help of inverse matrices.
• Elementary row operations and elementary matrices.  Elementary row operations method for finding inverse matrices.
• Equivalence of Ax = 0 only for x = 0 to the invertibility of A.

Euclidean spaces (Rⁿ)

• Operations in Euclidean spaces. Linear independence of vectors.  Subspaces. Bases. Dimensions of subspaces.
• Matrices as linear functions on sets of vectors (e.g. rotation, reflection)
• Eigenvalues and Eigenvectors

Determinants

• Permutations, inversions. Definition of the determinant of an n x n matrix.
• Properties of determinants. Minors and cofactors. Cofactor expansions.
• Equivalence of invertibility of matrix to non-vanishing of the determinant. Formula for inverse of a matrix. Cramer's rule.

Vectors and coordinate geometry

• Geometric and algebraic definitions of vectors.  Addition, subtraction, scalar multiplication, magnitudes, dot products and cross products of vectors, in geometric and algebraic forms.
• Equations of lines and planes.  Intersections of, or distances between, points, lines and planes.

See Background Reading List

Anton H., Elementary Linear Algebra, Wiley

Hamilton, A. G., A first course in Linear Algebra, CUP