may-8

These lecture notes were generated by Claude inside a Cowork environment with knowledge of the syllabus and course materials, and generated from the audio transcript of the lecture. The lecture notes are not a transcript, and skip some parts while adding small details with sources/clarifications. These notes have been checked by me to be correct. I have added and edited the file for correctness and alignment with necessary material. All right-margin aside notes are additional to the course, and are not part of the learning objectives.

Lecture 2 — May 8, 2026

Quercus outage

Quercus (Canvas) is currently down due to a security incident affecting Instructure’s platform globally. Until it is restored:

• Piazza is the primary communication channel for announcements and questions.
• All general course information is duplicated on the course website: denisgorodkov.com/mata22
• The first biweekly reflection is postponed to next week; a background poll (program, degree, math background) will go out instead.
• The Friday tutorial section is affected by the holiday — that section will likely not have a first tutorial; practice problems will still be posted. Details to follow.

Recommended resource

3Blue1Brown — Essence of Linear Algebra (YouTube playlist): a series of short, visually rich videos by Grant Sanderson. The treatment is less proof-based than our course, but the geometric intuition is excellent — especially for vectors and linear transformations. Highly recommended as a companion to the readings.

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1. Sets

A set is a collection of objects, called its elements. The useful mental model: a set is an answering machine — you can ask it “is an element of you?” and it always answers yes or no.

Notation

Elements are listed inside curly braces. The set above has exactly two elements: the number and a flower symbol. The symbol (read: belongs to, is an element of) records membership: , , .

The empty set has no elements — it always answers no. Note the subtlety:

The set has one element (which happens to itself be a set). This distinction matters: set theory can encode the natural numbers purely from , with , , , and so on.

A note on foundations

A fully rigorous definition of “set” requires an axiomatic system (Zermelo–Fraenkel). The naive definition — a set is any collection of objects you can describe — leads to contradictions (Russell’s paradox: the set of all sets that do not contain themselves). In this course, sets will always be well-behaved; we will never need to go near the foundations.

Further reading Naive Set Theory (1960). Chapter 1 takes about 20 minutes to read and resolves most of the foundational anxiety.

For a gentle and elegant account of naive set theory, see Paul Halmos,

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2. as a structured set

The real numbers are not just a set — they carry operations that relate elements to one another. The two fundamental operations are:

Addition: a function

Multiplication: a function

(The notation is introduced below.) Addition satisfies, among other things:

• Commutativity:
• Associativity:

These properties are not automatic — they are what makes particularly well-behaved, and they will reappear as axioms when we define vector spaces.

Beyond : the normed division algebras

The real numbers can be extended in a sequence of steps, each gaining something but losing a property of the arithmetic:

• (complex numbers): lose ordering, gain algebraic closure
• (quaternions): lose commutativity of multiplication
• (octonions): lose associativity of multiplication
• (sedenions): lose the division property

By a theorem of Hurwitz (1898), this list is exhaustive: the only normed division algebras over are . We will use meaningfully later in the course (eigenvalues, characteristic polynomials). See quaternions and octonions.

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3. Cartesian products and

The Cartesian product of two sets and is

Elements are ordered pairs — order matters: in general.

Taking iterated products of with itself:

An element of is an ordered -tuple of real numbers. Geometrically:

• — a line
• — the plane (every point identified by two coordinates)
• — three-dimensional space

Axler §1.A

Axler begins immediately with and , defining addition and scalar multiplication in exactly the form introduced today. Reading §1.A is the primary suggested follow-up for this lecture. (Free PDF)

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4. Why higher dimensions?

For most of the 19th century, mathematicians working in dimensions above 3 felt obliged to justify themselves. The modern answer is twofold.

Configuration spaces

To describe the position of an object in space, you need coordinates for its centre of mass (3 numbers) and for its orientation. A rigid body like a piece of chalk requires 5 parameters (or 6, depending on convention): 3 for position, and 2 angular coordinates for orientation (since orientation in lives on ). The set of all possible positions of such an object is a 5-dimensional configuration space.

Note

Configuration spaces are fundamental in classical mechanics and robotics. See Wikipedia: Configuration space.

Data

Each patient in a medical study might be described by ~15–20 numerical measurements from a blood test. Every person is then a point in . To find clusters, compute averages, or predict outcomes across a population, one needs geometry in . In machine learning, data points routinely live in spaces of thousands or hundreds of thousands of dimensions.

Geometric intuition in higher dimensions

Direct geometric vision does not extend past . What does extend is the logical structure of geometry. Example:

• In : two planes either coincide, are parallel, or intersect in a line.
• In : two planes (each 2-dimensional) can intersect in a single point — the fourth dimension provides the extra room for the planes to “tilt away” from each other in both dimensions simultaneously.

The goal over the semester is to identify which geometric intuitions transfer to and which do not.

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5. Vectors in

A vector in is simply an element of . We write it as a column:

Alternative notations: , , . The column convention will become natural once we multiply matrices by vectors.

The real numbers are called the components (or coordinates) of .

Hefferon Ch. 1, §II.1

Hefferon introduces vectors geometrically in Chapter One, §II.1 (“Vectors in Space”), motivating them as free displacements before defining and componentwise operations. The formal vector space definition comes later in Chapter Two. (Free PDF)

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6. Operations on vectors

Addition

Vector addition is defined componentwise, inheriting its definition — and its properties (commutativity, associativity) — from the addition in .

Geometrically in : is the diagonal of the parallelogram spanned by and (parallelogram law).

Scalar multiplication

Real numbers, when acting on vectors, are called scalars — because their role is to scale vectors. For :

Geometrically: stretches by a factor of and reverses its direction if .

Why Greek letters for scalars?

Convention, not mathematics. Greek letters () are traditionally used for scalars in linear algebra so that a formula like can be scanned quickly — Greek = scalar, bold/arrow = vector. You will not be penalized for notation choices in the first several weeks (and probably not penalized at all if your solutions stay readable).

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Looking ahead

Next week’s tutorials will focus on working concretely with vectors in : computing sums, scalar multiples, and interpreting them geometrically. Sincerity of approach matters more than correctness for Tutorial 1 — show your thinking, including where you get stuck. Some tutorial problems will also go deeper in set theory: discovering operations on sets and their properties.

The abstract definition of a vector space — generalizing to arbitrary sets equipped with addition and scalar multiplication satisfying certain axioms — will appear next week. Today’s operations and their properties (commutativity, associativity, distributivity) are exactly the axioms we will formalize.

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Textbook references

Note on Axler edition

The references uses the 2nd edition of Axler, which has a different chapter structure from the widely-cited 4th edition. In the 2nd edition, Chapter 1 has no §1.A/§1.B subsection labels — sections are titled “Complex Numbers”, “Definition of Vector Space”, etc. I will try to later switch the notations to the 4th edition.

Topic	Axler (2nd ed.)	Hefferon (4th ed.)	Treil
, vectors, components	Ch. 1, “Definition of Vector Space”	Ch. 1, §II.1	Ch. 1, §1.1 (Example 1)
Vector addition	Ch. 1, eq. 1.1	Ch. 1, §II.1	Ch. 1, §1.1
Scalar multiplication	Ch. 1, “Definition of Vector Space”	Ch. 1, §II.1	Ch. 1, §1.1
Vector space axioms (preview)	Ch. 1, “Definition of Vector Space”	Ch. 2, §I.1, Definition 1.1	Ch. 1, §1
Complex numbers / scalars	Ch. 1, “Complex Numbers”	—	Ch. 1, §1 (remark on )