These lecture notes were generated by Claude inside a Cowork environment with knowledge of the syllabus and course materials, combining the blackboard photos from lecture with student notes. 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 1 — May 6, 2026
1. What is linear algebra?
The name has two parts.
Linear means a constant rate of change between two variables. The prototypical linear equation is
where
Contrast this with an equation like
which is non-linear: the rate of change depends on where you are.
Algebra, in the sense used in higher mathematics, is the study of structures — sets equipped with operations that satisfy specified rules. This is different from the high-school sense of algebra (solving for
Together, linear algebra is the study of linear transformations, vectors, and vector spaces — structures where you can add elements and scale them, subject to a clean list of axioms.
Why linearity matters
A useful heuristic: zoom in closely enough on any smooth curve, and it eventually looks like a straight line. Linearity is not just one special case among many — it is the local approximation to all of smooth mathematics. This is why linear algebra appears everywhere, from physics to machine learning to differential equations.
2. Points and vectors
We begin with two kinds of geometric object that are easy to confuse, but which behave very differently.
A point is an exact location in space. It has no size, no width, no direction — it simply is somewhere. A line contains infinitely many points. Points answer the question: where?
A vector has both a direction and a magnitude (length). It is drawn as an arrow: the tail is where it starts, and the tip (the arrowhead) is where it ends. Vectors answer the question: which way, and how far?
The campus example
To make the distinction concrete, consider three University of Toronto campuses: UTSC, UTM, and UTSG. Placed in a coordinate plane, they might sit at positions
As points, these are just locations. The distance from the origin to UTSG is
As vectors, we can draw an arrow from one campus to another — for example, from UTM to UTSG. That arrow encodes the displacement: how far, and in what direction, you would have to travel. The vector does not care where on the page it is drawn; two arrows with the same length and direction are the same vector.
The distinction: the cities are points — fixed locations. The directions and distances between them are vectors.
Affine spaces
Mathematicians have a name for a space where you have points without a preferred origin: an affine space. In an affine space, you can subtract two points to get a vector (the displacement between them), but you cannot add two points and get a meaningful answer. In
there is a canonical origin , which lets us quietly identify points with vectors by measuring displacements from . This identification is so natural that we usually forget it is a choice. It will fail in curved spaces.
3. Vectors in the real world
Vectors from physics — arrows that can be added by placing them tip to tail — work cleanly in a flat plane or in flat three-dimensional space. In the real world, complications arise.
The North Pole
Where on Earth would vectors using the cardinal directions not work? At the poles. Standing at the North Pole, there is no “north” direction to move in — you are already as far north as possible. Every direction from the North Pole is south. The standard compass-direction vectors simply break down.
Order matters on a sphere
On a flat plane, if you travel 1 km east and then 1 km south, you arrive at exactly the same point as if you had gone 1 km south first and then 1 km east. Vector addition is commutative on flat space.
On the surface of the Earth (a sphere), this fails. Travel 1 km south from a point on the equator, then 1 km east — you arrive at a different location than if you had gone east first and then south. On a curved surface, the order of travel matters.
This is a preview of a deep idea: the geometry of vectors is sensitive to the underlying space. The clean rules of linear algebra hold in flat spaces (
Tangent spaces
The right way to do calculus on a sphere is to attach a separate flat vector space to each point — the tangent plane at that point. At the North Pole, it is the plane that touches the sphere from the top; at a point on the equator, it is a different, tilted plane. Every point carries its own vector space. These tangent spaces fit together into a structure called a tangent bundle. This is the beginning of differential geometry and general relativity.
Looking ahead
The next lecture will develop the formal machinery needed to make today’s ideas precise: what exactly is a set, what structure does
The abstract definition of a vector space — which generalizes
Textbook references
| Topic | Axler (2nd ed.) | Hefferon (4th ed.) | Treil |
|---|---|---|---|
| What is linear algebra? | Ch. 1 (intro) | Ch. 1, §I.1 | Ch. 1, intro |
| Points vs. vectors | — | Ch. 1, §II.1 | Ch. 1, §1.1 |
| Vectors in | Ch. 1, “Definition of Vector Space” | Ch. 1, §II.1 | Ch. 1, §1.1 |
| 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 |