MAT185: Linear Algebra

Why were the Wright Brothers linearly independent vectors?
Because two of them made a plane!

[Source]

Linear Algebra is a basic field of math that is used in all sorts of engineering and science fields. In fact, the first step in solving many engineering problems is to make it a linear algebra problem. It’s no surprise that almost all engineering and science programs teach linear algebra very early on.

MAT185 is loosely a continuation of ESC103. It teaches Linear Algebra from a first principles, ground-up approach. You will learn the reasoning behind mathematical ideas and rigorously prove that they are true. You will cover concepts like vectors, matrices, eigenvalues, and eigenvectors that were introduced in ESC103, as well as some new concepts like fields, vector spaces and bases.

However, unlike ESC103, there is little-to-no computation in this course. This is a proof-based course, so you will be tested on your ability to connect concepts, logical reasoning, and proving and disproving general statements. MAT185 is taught as if you have never taken a proof-based course before, so don’t worry if you are new to this: it’s time to learn! Students have varying experiences with this course. Some find it reasonable while others find it very difficult. There is little correlation between how you felt about ESC103 and how you will feel about MAT185. Although they both cover Linear Algebra, the questions you are asked, the perspective from which you learn, and what you are expected to understand are completely different.


Professors

MAT185 will be taught by Professor Sean Uppal and Professor Philipp Seieler.

Professor Philipp Seiler

Professor Philipp Seiler [Source]

Professor Seiler completed his Bachelors in Mechanical Engineering and Mechatronics from TU Braunschweig, in Germany. He then completed his PhD in Material Science. He recently transferred to be a Professor at U of T, but has been teaching for many years as an assistant professor at the University of Kent in England. His research focuses on materials under extreme conditions, such as materials for rockets engines or gas turbines and lightweight materials.

Professor Sean Uppal

Professor Sean Uppal [Source]

Professor Uppal is well-known in the mathematics department at U of T, and he taught this course last year. He has some great stories (one stands out in particular), but you’ll have to ask him if you want to hear them.

Professor Interview

Interview Transcript


Highlights

  • Not only will you learn new theorems, but you will learn how to prove them so that you know they are true!

  • A lot of math symbols. (Don’t worry, the professors will walk you through them.)

Week in the Life of an MAT185 Student

Lectures

There are three hours of lecture per week in this course. Lectures cover proofs, explanations of theorems, and concepts. The professors are usually very clear with their writing on the board, and everything they write down will be relevant. Take notes. There are minimal worked examples in lecture so make sure you practice in tutorials or on your own time. If you have questions, don’t hesitate to ask the professors after class.

Tutorials

There is one hour a week of tutorials in this course. In tutorials, you will be given practice problems and the TA will help you solve them. This is a useful tutorial where you can get a lot of practice, but it is only as good as you make it.

Assignments

Approximately every month there is a problem set consisting of 3 questions. You are usually asked to prove or disprove a statement. The problems are relatively difficult, but you get a week to think about them and work on a solution. Try your best and these problem sets will be valuable practice! The problem set contents are also similar to the more difficult exam questions. 

In addition to these problem sets, the professors will give you a list of recommended problems for every week. Do these. They are not marked and are technically optional, but you will not succeed in this course without them. If you can’t complete all of these recommended problems, solve as many as you can.

Exams

This course has two midterms and a final exam. To study for these assessments, find 3-5 past exams and work through them. Once you understand how to think about problems in this course and have seen sample solutions, you will begin to adopt the right problem-solving mindset.

Note that you will not succeed in this course just by completing a few past exams. You need to practice regularly in order to build your problem-solving skills.

Find past MAT18 Exams on courses.skule.ca.


How to Succeed

Quick tips and equations

  • Understand the concept of vector spaces. As you’ll learn soon enough, vectors are more than just “pointy arrow thingies!” Know the proof for vector spaces by heart.

  • Know the difference between \bigcap and \bigcup , as well as \subseteq and \subset .

  • Thinking visually! (This is a great resource to help understand Linear Algebra visually.)

  • \text{dim}(\text{null}(\textbf{A})) = n - \text{rank}(\textbf{A}) , where \textbf{A} is a n \times m matrix. This is called the fundamental theorem of linear algebra and is one of the most important concepts in linear algebra. It states that the dimension of the null space of a n \times m sized matrix is equal to n – the rank of that matrix! Don’t worry if you are unfamiliar with these terms you will learn them in MAT185!

  • \text{rank}(\textbf{A}) = \text{dim}(\text{col}(\textbf{A})) = \text{dim}(\text{row}(\textbf{A}))

  • \text{det}(\textbf{A}) - \lambda\textbf{I} = \textbf{0} : Characteristic equation of matrix \textbf{A}

More Details

Practice, practice, practice

As mentioned earlier, practice is necessary for success in MAT185. Solving a variety of problems will help you learn different problem-solving methods. You will become more comfortable with proofs and will build your linear algebra intuition – both critical in this course.

Don’t Cheat Yourself with Solutions

Sometimes, it’s difficult to even start a problem in Linear Algebra. Don’t cave in and look at an answer key right away: this practice will hurt you in the long run.

If you don’t know how to start the problem, write down what you know about the problem instead – things such as relevant equations, facts, and theorems. Once all your tools are laid out in front of you, it is easier to connect the dots and come up with a solution!

Have a Concise Collection of the Theorems and Proofs

In this course, your main job is connecting different facts and theorems to prove and disprove statements. It’s a good idea to have a cheat sheet where you’ve written down the main theorems for quick reference. Physically organizing and writing down theorems and equations will also help you get them straight in your head and understand how they connect.

Don’t Ignore the Review at the Start of the Course

This course starts slowly, with review from ESC103 in order to build a strong foundation for later material. Pay attention and work hard at the start of this course, and you will reap the benefits later down the line. All the content in MAT185 is VERY connected, so a shaky foundation will make your semester hard later.

Make Use of Other Resources

Linear Algebra is not a new subject. There are a lot of online videos and tools to help you learn it. If you have trouble understanding a concept, look online for another perspective on it. The great thing about Linear Algebra is there are several ways of understanding it. Sometimes a physical, geometric interpretation makes the most sense. Other times, equations will just work in your head. Experiment and find out what’s best for you.


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; there are often many ways to approach a problem, and you’ll get to experiment with this.

  • Linear Algebra is applied all over engineering and science. For example, most circuit problems are solved using matrices. In physics, quantum mechanics makes use of special matrices to determine what is possible for a particle. In computer science, vectors can be used in gaming and graphics. You will be using a lot of linear algebra principles for working in artificial intelligence, robotics and machine learning. Linear algebra is a useful and necessary tool for any field you’re interested in.