Why were the Wright Brothers linearly independent vectors?
Because two of them made a plane!
Linear algebra is a field of math that is used in many 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 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 some concepts that were introduced in ESC103, such as vectors, matrices, and differential equations, and many new concepts including fields, vector spaces, bases, coordinates, linear transformations, and eigenproblems.
However, unlike ESC103, there is little computation in this course. This is a proof-based course, so you will be tested on your ability to connect concepts and use linear algebra principles to prove and disprove 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 some pure math and linear algebra, the questions you are asked, the perspective from which you learn, and what you are expected to understand are completely different.
Professor
Professor Philipp Seiler
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 rocket engines or gas turbines and lightweight materials.
Professor Interview
Snippets from our interview with Professor Seiler:
“In my research, […] I don’t think there is a single day where I don’t use linear algebra.”
“I have an active teaching style. I like going on academic detours […] I do not plan lectures down to the minute. I come to lectures with a brief outline, then I see how the students follow and react.”
“One by one, I will introduce the necessary concepts for [students] to understand [linear algebra] […] If I were to simply teach how a tool like matrix multiplication works, e.g. just by following an algorithm, students would quickly forget it. By knowing the proofs, you can understand why and how matrix operations work. Even if you forget the details, it will be easier to relearn it. Moreover, I want students to understand why certain concepts are true and why they work. Therefore, proofs can help students understand linear algebra conceptually.”
For more, check out the Interview Transcript here.
Course Highlights
- The course textbook. We don’t want to spoil your fun, so read it for yourself!
- 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.)
- The pure satisfaction you gain from proving difficult mathematical statements by using fundamental linear algebra concepts.
Week in the Life of an MAT185 Student
Classes
There are three hours of lecture per week in this course. Lectures cover proofs, explanations of theorems, and concepts. The professor is usually very clear with his writing on the board, and everything he writes will be relevant, so take notes. There will be some examples, including graphical explanations and engineering applications to linear algebra, but ensure you do additional practice in tutorials or on your own time. If you have questions, don’t hesitate to ask the professors after class.
Before every lecture, you will have to complete a textbook reading along with an online quiz. Ensure that you do these, as they are crucial to understanding the concepts that will be covered more in-depth during the lecture (they are also worth marks)!
There is 1 hour of tutorials per week 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.
Assessments
Approximately every month there is a problem set consisting of 3 questions. You are usually asked to prove or disprove some statements. 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.
This course has two midterms and a final exam. To study for these assessments, complete many homework and tutorial problems, and past exams. 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 and develop intuition as to when you should apply certain linear algebra principles. Note that you will not succeed in this course just by completing a few past exams. You will need to practice regularly and truly attempt to digest all of the content.
Find past MAT185 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{rank}(\textbf{A}) = \text{dim}(\text{col}(\textbf{A})) = \text{dim}(\text{row}(\textbf{A}))
- The rank-nullity theorem: \text{dim}(\text{null}(\textbf{A})) = n - \text{rank}(\textbf{A}) , where \textbf{A} is an m x n matrix with real values.
- \text{det}(\lambda\textbf{I} - \textbf{A}) = \textbf{0} : Characteristic equation of matrix \textbf{A}
More Details
MAT185 builds upon concepts from ESC103 such as vectors and matrices and requires you to use them for proofs instead of computations. Therefore, you should thoroughly understand all the content from ESC103; concepts in MAT185 are VERY connected, so a shaky foundation will make your semester more difficult later.
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.
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 a problem, write down what you know about it, such as relevant equations, facts, and theorems. Once these tools are laid out in front of you, it’ll be easier to connect the dots and develop a solution.
Even if you do think that you know how to solve a problem, ensure that you can solve it with a formal and rigorous proof! That being said, do not waste your time creating a very sophisticated proof for every single easy question.
In this course, your main job is connecting different facts and theorems to prove and disprove statements. Physically organizing and writing down theorems and equations will help you get organized in your head and understand how they connect.
Linear algebra is not a new subject. If you have trouble understanding a concept, there are a lot of online resources through which you can gain intuition and look for different perspectives. These different interpretations are what make linear algebra great: sometimes a physical, geometric interpretation makes the most sense. Other times, equations will just congregate together in your head. Experiment and find out what’s best for you.
Beyond First Year
- 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 has applications all over engineering and science. For example, most circuit problems are solved using matrices. Quantum mechanics make use of special matrices to determine what is possible for a particle. In computer science, vectors can be used in gaming and graphics. Google uses eigenvectors to determine the ranking of pages in a search. Linear algebra is a necessary tool for robotics, machine learning, and for any field you’re interested in.
- Many of your upper-year courses will require strong knowledge and frequent usage of linear algebra!