My brave warriors and adventurers, welcome back. It’s time to orient ourselves with the eastern bank of the River of Proof and take a glimpse into the land of algebra. The land can be split into two main areas: linear algebra and abstract algebra.
Does the rocky cliff look familiar to you? That’s the Matrix Cliff! Adventurers who come into this land are likely to already know about matrices – rectangular arrays of numbers or symbols. It is of no doubt that you will repeatedly encounter such cliffs: matrices are very useful tools in representing and operating multiple linear equations and linear transformations, which lie at the centre of the study of linear algebra.
You may feel like climbing up the cliff to breathe in some fresh air and enjoy a birds-eye view of the landscape. In doing so you will see the vast space spreading out in front of you, which is given the name ‘Vector Space’. A vector space, to be understood simply, is a set of mathematical objects which can be added up and multiplied by numbers, and most importantly, whose additions and multiplications have to satisfy a few specified axioms (more information can be found here). After a vector space is defined, we call its elements ‘vectors’. Do remember, a vector is defined to be an element of a vector space, and it can be more than an arrow or a string of numbers like the column of a matrix. In fact, polynomials are vectors in a polynomial vector space, and functions are vectors in a function vector space. The common theme is that the set which these vectors belong to all satisfy the vector space axioms.
When wandering across the land, be sure to pay a kind visit to the Wizard of Linear Transformation, and try your best to get on his good side! Basically, a linear transformation T is a map from a vector space V to a vector space W that … does what the wizard’s spell says. If you are lucky enough to have the wizard teach you some of his magic, you may be interested to learn the concepts of images, kernels, rank, nullity and the very essential Rank-Nullity Theorem which ties them all together.
In this area you should also explore the nearby inner product space, which is a special kind of vector space. There you will learn about orthogonal matrices and prove the spectral theorem, revealing its close relationship with rotations and reflections – themselves important concepts in geometry.
Abstract Algebra – Groups
Turning to the other side of the land of algebra, your gaze will no doubt be caught by the idyllic scenery of abstract algebra: fields, groups of maples, rings of oaks … everything looks so well-structured and in order, making people wonder of the mystery that lies within. Groups, fields, rings, modules, vector spaces, all of these belong to the family of algebraic structures, and abstract algebra is the study of these algebraic structures.
You will start your trip from the Woodland of Groups. Many interesting events take place there and I hope you have fun without straying too far from the beaten path. For each set of maple (or numbers, rotations, whatever) to be a group, it is necessary to have an operation (examples of binary operations are matrix multiplication, function composition, addition…) over the set that satisfies the group axioms (a list of rules not too dissimilar to the vector space axioms mentioned above). Some special groups you will explore include permutation groups, dihedral groups (the group of maps from a polygon into itself otherwise known as its symmetries) and rotational groups. At first study of these groups may seem as abstract as it sounds, but actually, the knowledge can be applied to solve questions like ‘How many distinct ways are there to make a bracelet with one red bead, two yellow beads and three blue beads?’, ‘How many distinct designs are there to colour the faces of my cubic box with black and gold?’ and many more besides.
Within groups you will be able to identify structures called subgroups. The relations between a group and its subgroup resemble those between a vector space and its subspace. An undoubted milestone of your adventure will be to successfully prove Lagrange’s Theorem, which concerns the cardinalities of a group and its subgroups. Other essential concepts to learn about are homomorphisms and isomorphisms, with the latter giving rise to the very interesting fact that two groups which seem apparently different can be identical in essence.
Around the Woodland of Group Theory, the Mountains of Number Theory twist and twine, stretching endlessly. Number theory, the study of integers, indeed contains many sheer peaks difficult to conquer. For example, generations of great adventures have devoted their lives to conquer the Peak of Fermat’s last Theorem. And eventually, Sir Andrew Wiles became the first person to stand on the peak to witness the most gorgeous sunrise. However, many nearby peaks, such as the Peak of Goldbach’s Conjecture sitting atop the neighbouring Mt. Prime Numbers, remain covered with clouds…
As a first-year warrior you will be able to prove Fermat’s little theorem and the Chinese remainder theorem using the methods of groups, hinting at the many underlying links between group theory and number theory that will be uncovered further along your journey.
In Part 3 of the Adventurer’s Guide to Mathematopia (coming soon) we will continue our journey to the remaining parts of the map: the Land of Geometry and Probability Place.
The full ‘Map of Mathematopia’ can be found here.
Part 1 of the Adventurer’s Guide to Mathematopia can be found here.