Linear algebra and calculus are the two most important foundational pillars on which modern mathematics is built. They are studied by almost all mathematics students at university, though typically labelled as different subjects and taught in parallel. Over time, students discover that linear algebra and calculus are inseparable (but not identical) twins that interlock to form the backbone of almost all applications of mathematics to physical and biological sciences, engineering and computer science. It is recommended that participants in the MOOC Introduction to Linear Algebra have already taken, or take in parallel, the MOOC Introduction to Calculus.
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All of our modern technical and electronic systems, such as the internet and search engines, on which we rely and tend to take for granted in our daily lives, work because of methods and techniques adapted from classical linear algebra. The key ideas involve vector and matrix arithmetic as well as clever methods for working around or overcoming difficulties, a form of obstacle avoidance, articulated in this course as the Conjugation Principle.
This course emphasises geometric intuition, gradually introducing abstraction and algebraic and symbolic manipulation, while at the same time striking a balance between theory and application, leading to a mastery of key threshold concepts in foundational mathematics.
Students taking Introduction to Linear Algebra will:
• gain familiarity with the arithmetic of geometric vectors, which may be thought of as directed line segments that can move about freely in space, and can be combined in different ways, using vector addition, scalar multiplication and two types of multiplication, the dot and cross product, related to projections and orthogonality (first week),
• develop fluency with lines and planes in space, represented by vector and Cartesian equations, and learn how to solve systems of equations, using the method of Gaussian elimination and introduction of parameters, using fields of real numbers and modular arithmetic with respect to a prime number (second week),
• be introduced to and gain familiarity with matrix arithmetic, matrix inverses, the role of elementary matrices and their relationships with matrix inversion and systems of equations, calculations and theory involving determinants (third week),
• be introduced to the theory of eigenvalues and eigenvectors, how they are found or approximated, and their role in diagonalisation of matrices (fourth week),
• see applications to simple Markov processes and stochastic matrices, and an introduction to linear transformations, illustrated using dilation, rotation and reflection matrices (fourth week),
• see a brief introduction to the arithmetic of complex numbers and discussion of the Fundamental Theorem of Algebra (fourth week).
Syllabus
Week 1 - Geometric Vectors in the Plane and in Space
This module introduces and explores the useful and elegant arithmetic of geometric vectors, regarded as directed line segments, which move about freely in the plane and in space. Objects in this arithmetic are ubiquitous throughout the physical world, modelling vector quantities. Students acquire tools that enable them to explore precise geometrical relationships between objects, prove difficult theorems and solve optimisation problems. This arithmetic forms a prototype for the general and abstract theory of vector spaces, developing students’ intuition and preparing them for advanced courses on linear algebra.
Week 2 - Lines and Planes in Space and Systems of Linear Equations
This module introduces and develops fluency with lines and planes in space, represented by vector and Cartesian equations. Students learn how to exploit the arithmetic of geometric vectors to solve difficult optimisation problems such as finding the closest point to a plane or finding the closest points on a pair of skew lines. The module introduces systems of linear equations and then develops the method of Gaussian elimination, using elementary row operations, followed by back substitution, to express solutions in terms of parameters. Systems of equations are also explored and solved using modular arithmetic with respect to a given prime number.
Week 3 - Matrix Arithmetic and the Theory of Determinants
This module introduces matrix arithmetic and the theory of determinants. Students first learn how to add matrices of the same size and how to multiply by a scalar. They then learn how to multiply matrices of compatible sizes, using cascades of dot products of rows with columns. This is described succinctly using Sigma notation, which is then used to prove associativity of matrix multiplication. Students learn about elementary matrices, which are basic building blocks in matrix arithmetic, closely related to elementary row operations used in Gaussian elimination. Students learn about matrix inverses and how to find them. Students learn about determinants and their properties, including the multiplicative property and a simple criterion for recognising invertibility of a matrix. Students see applications to cross products of vectors and for exploring spatial relationships between points and triangles.
Week 4 - Eigentheory and Diagonalisation
This module continues the development of matrix arithmetic by introducing eigenvalues and associated eigenvectors. Interpreted geometrically, these allow one to find directions in which a given linear operator associated with a matrix moves vectors in straight lines. This leads to the technique of diagonalisation, enabling one to solve difficult problems in matrix arithmetic, including finding formulae for powers of a given square matrix. This has many applications, including, for example, in exploring the behaviour of a Markov process described by a stochastic matrix, the mathematics of which underlies search engines on the internet. Eigenvalues can be found, in principle, by solving the characteristic equation of a matrix. Associated eigenvectors can then be found by solving an associated homogeneous system of equations. In practice, there are iterative numerical techniques for finding approximations of eigenvalues and eigenvectors, using a technique associated with Perron’s Theorem. Diagonalisation is a manifestation of the general Conjugation Principle, explored in different contexts. Linear transformations are introduced, focusing on transformation of the plane. Rotations and reflections of the plane combine to form the two-dimensional orthogonal group. Scalar dilations and rotations combine to form a copy of the field of complex numbers. A sketch of Smale’s proof of the Fundamental Theorem of Algebra is given, which says that any nonconstant polynomial with complex coefficients has a complex root, so that all square matrices have eigenvalues, when working over the field of complex numbers.
