Linear algebra is a basic field of math that is used in all sorts of engineering and science fields. In fact, one of the first steps in solving a lot of problems is trying to make it a linear algebra problem. So it’s no surprise that all engineering and science programs teach linear algebra very early on.

MAT185 is a continuation of ESC103. Like ESC103, MAT185 is a pure math course, you learn mathematical concepts without connecting them much to other fields. In the course you’ll review vectors, matrixes, and eigen values and eigen vectors. You then move to deeper concepts like basis and vector spaces, things you probably haven’t seen in highschool or ESC103.

The questions in this course will often require you to use the principles taught in class to prove or disprove something. Computation is not a focus of this course, there are very few questions that ask you to calculate something.

Students have varying experiences with this course, some find it reasonable while others find it very difficult. You shouldn’t think that your experience with ESC103 will be the same as this course because the material is more advanced and sometimes requires a completely different perspective.

**Professor**

You will likely have Professor G. M. T. D’Eleuterio, who works at the University of Toronto Institute for Aerospace Studies (UTIAS). He has been teaching this course for more than 15 years; he even wrote the textbook that you all will be using for this course! Professor D’Eleuterio is also an EngSci graduate.

**Key Highlights**

- The textbook!! MAT185 has one of the most interesting textbooks you will read. It’s called “An ALGEBRA PROFESSOR in the HOUSE of the MEDICI”, and is written by J. W. Lorimer and G. M. T. D’Eleuterio (Professor D’Eleuterio will likely be teaching you the course). The textbook is a time travel story about a math professor teaching an Italian nobleman about math; it sounds strange but it’s a great read!

Here is the cover page of the textbook:

- You will learn new theories that you can use as tools to solve problems. You will learn a lot about using these tools to solve a proof. So pay attention!

- A lot of math symbols. More info here

**What will you take out of it?**

- This course will give you lots of problem solving experience! Linear algebra is a very abstract and general topic in math, so there are often a lot of ways to approach a problem. You’ll get some experience in knowing when to think about a problem geometrically, as an equation, or in a new way unique to the problem.

- Linear algebra is applied all over engineering and science. For example, most circuits problems are solved by placing the equations in a matrix. In physics, quantum mechanics makes use of special matrices to determine what is possible for a particle. In computer science vectors can be used to store any collection of data. You will be using a lot of linear algebra principles for working in Artificial Intelligence, Robotics and/or Machine Learning as well. Linear algebra is a useful and necessary tool for any field you’re interested in.

**Some tools for success**

- MATLAB
- Trace, Rank, and Spread of a matrix
- Vector Spaces

**How to succeed? **

**Don’t cheat yourself with solutions**

- You’ll often feel that you have no way to start tackling a problem, and be tempted to look at the solutions. Obviously, this is a really bad idea because you won’t have solutions on the midterm. There aren’t a lot of problems on the problem sets so wasting them by checking the solutions will weaken your problem solving ability.

- What you should do when you have no idea how to start a problem is to think about the topic the question is most related to and then write down the most important facts and theorems about that topic. Once you can see the facts in front of you you’ll probably see a way to connect them and move forward on the question.

**Have a concise collection of the theorems and proofs**

In this course your main job is connecting different facts and theorems to prove statements. It’s a good idea to have a cheat sheet where you’ve written down the main theorems so you have a quick reference when practicing. Organizing the collection of theorems and facts will also help you remember them and see how they connect.

**Make use of other resources**

Linear algebra is a very old and standard course, so there are a lot of online videos and tools for learning it. If you have trouble understanding a concept, look online for another perspective on it. One great thing about linear algebra is that there are lots of ways to learn it, sometimes a geometric analogy is best while other times an equation standpoint is preferred.

**Don’t ignore the review at the start of the course**

For the first couple weeks of the course there will be a lot of review from ESC103. This may give you the impression that the course is very easy. Obviously this is not the case; the review will be from a slightly more detailed perspective and will quickly lead to more complex topics. Remember to take notes and pay attention to save some headaches later on.