Advice from former students for MTH 2775 - Linear Algebra

• From Brandon Amann (Major: Mechanical Engineering, Fall 19)

Me: What is your advice to prospective students to be successful in this course?

Brandon: To be successful in MTH 2775 with Professor Tran it is imperative that you practice the material on your own and utilize the resources provided by Professor Tran. As I am sure you are aware, familiarizing yourself with the homework problems is the best advice anyone can give. For me, it was important that I did the homework problems after each lecture to enable myself to ask any questions I may have during the following lecture. One thing that I find to be more helpful is to do the homework on a day other than the day you learned it. This allows for students that don't know the solution to a problem to review their notes from the prior session and rebuild their knowledge of that material.

When it comes to studying for the test, there are a few things that I did to help myself prepare. To start, once all the material was covered, I rewrote my notes for each section to re-familiarize myself with the necessary theorems, definitions or important examples. It is important to know the theorems behind each concept and not any shortcuts that may be taught to simplify the workload. Once completed, I then began working on answering the questions that Professor Tran provides as a sample exam. Although this is an excellent way to prepare, the more problems you do the better. For this reason, I selected a handful of problems within the homework, or similar to it, that accurately represented the material on the exam. I then shuffled the problems from the textbook so that I did not know the way to work out a solution based on the section I took it from. This proved to be helpful for me to be able to recognize how to solve any problem.

My last piece of advice would be for students who are stressing during the exam. One thing that helps to remember if you are struggling to solve a problem is that, often times, the different parts of a problem are related. For example, if part (a) of a problem asks to state a theorem and part (b) asks you to prove something, then more than likely you will utilize that theorem in your proof. Although a simple testing tactic, it proves beneficial if you are stuck on a multi-part problem.

Me: Which sections are most difficult? How do you study them?

Brandon: Many sections rely on knowledge from the other sections in order to fully understand the material. For this course, it is imperative that students are able to demonstrate their ability to solve matrices. Some of the more difficult concepts for me to grasp were the following: Row Space and Column Space (Image especially), Matrix representations for linear transformations (going from a basis in R2 to a basis in R3), and understanding the notion of change of basis and coordinates (understanding the notation of [v] with respect to a basis). To conquer these challenges, the best thing to do is to practice more problems than just the homework suggested. Familiarize yourself with the process of how to solve problems of this type. For example, rather than try to memorize the equation for a change of basis transition matrix, familiarize yourself with how that matrix is obtained through simple matrix algebra. If practicing representing a linear operation by a matrix, try to solve it without utilizing the complicated shortcut way to truly understand what you are doing. The only way to improve is to practice. Utilize your notes, your textbook, and your professor! They will all help you to pass this course!

• From Daphne Agapiou (High School Student, Winter 2020)

Me: How to be successful in MTH 2775 (with an A)? What is your advice to prospective students in this course?

Daphne: In order to be successful in Linear Algebra, it is important to understand the theorems, definitions, and main concepts. This class is not as calculation based, but more about proving different concepts. With this being said, you want to make sure you fully understand the theorems so you can apply it to different proofs. Instead of memorizing equations or how to solve a problem, as you would in calculus, you want to make sure to understand what the theorems mean because every proof is different.

In addition, it is important to show up to class and take notes. Pay attention to when Professor Tran makes “Remarks” because these are important concepts and will help you solve the problems in a section. Also, it helped me to do the assigned homework before the next class because Professor Tran will take time at the beginning of class to answer a few questions. Also, doing homework ahead will most likely help you better understand the concepts being taught in the next class because many proofs and ideas build on each other. As in other math classes, a concept you learned from the first few lessons could be used to help you solve a proof in more advanced lessons. For this reason, as the class progressed, it helped me to make a formula sheet of the different definitions and key concepts that were in notes and showed up in proofs. This way I was not overwhelmed by the material.

Lastly, it is very important to do the homework. While the internet can help give you hints where to start or what theorems might be useful in solving certain problems, it will not help you to merely copy answers from online because you will not understand the concepts and thought processes behind solving the problem or proving an idea. Make sure to prove/solve the problem on your own because Professor Tran often puts problems on his exam very similar to those in the homework so you will want to fully understand the thought processes behind a problem. Also, be sure to complete the practice exams before review days because you can ask questions and the actual exams will not be more difficult than the practice ones. The practice exams gave me a good idea of what lessons and concepts I did not have a full understanding of, so I would go back and read those lessons and even redo problems from the homework. If there is a concept or problem you do not understand, Professor Tran is extremely helpful. I went to his office hours whenever I had trouble with a homework problem or concept and he was always able to help give clarification or go through the steps in how to solve a problem.

Me: Which sections are most difficult? How do you study them?

Daphne: For me, the hardest sections were 3.4: Basis and Dimension, 3.6: Row Space and Column Space, and 5.2: Orthogonal Subspaces. The most important part of the class is having a full understanding of the theorems and how to apply them to problems. Although the concepts were difficult for me at first, it helped me to read the lessons in the book and go over the examples in the book and the examples Professor Tran provides in his lecture notes. This gave me an idea as to how I could approach certain problems. I would use a sheet I made with different definitions and theorems I learned throughout the sections because when you need to prove something, you want to start with a definition. Also, always ask questions in class or office hours if you are having trouble with something.

• From S. Avrusky (High school student, Winter 2020)

Me: How to be successful in MTH 2775 (with an A)? What is your advice to prospective students in this course?

S.: It’s very easy to get lost and left behind without even realizing it. The class starts off fairly simple, reviewing things from high school, but don’t let that deceive you. Before you know it, it has moved on to newer and harder material. My main advice is look out for yourself and realize when you are struggling. If you find yourself having to look up solutions to every single homework problem or you have no clue how to attempt the questions on the first quiz, it’s important to step back and fill in the gaps in your knowledge before moving on. Seek out help when you need it. Professor Tran is very open in his office hours and can give you immeasurable help. When you’re doing the homework, make sure you understand how each step flows into the next and you’re not just plug-and-chugging your way through the problem or mindlessly copying solutions off Slader. Pay attention during lecture to the things that maybe don’t make sense to you, so you know what to return and study again later. Plus, Professor Tran gives many hints and remarks during lecture that will be helpful on the exams. This class may be difficult, but it is totally possible to succeed in it if you put in the time.

Me: Which sections are most difficult? How do you study them?

S.: The class moves quickly and many sections build on previous lessons, and if you’re still struggling with concepts from the first few lectures, the later sections will be even harder. I found sections 3.2 Subspaces and 3.3 Linear Dependence to be the first substantial increase in difficulty. I think this is when I started learning new material instead of reviewing stuff from high school. Later, the proofs in sections 3.6 Row / Column Space and 5.5 Orthogonality were difficult for me too. Though, having taken APM 2663 beforehand, I had some experience with proofs to help me. With proofs, I find it helpful to write out what information I have to start with and what conclusion I want to arrive at. Even if you have to look up the solution, make sure you understand each step before you move on. Personally, I liked to preview the section in the textbook before coming to class, so I could follow along easier during the actual lecture. Homework and the sample exam questions are great study tools, too. They help you see what you do and don’t know and give good practice for the actual exams. Another strategy that I found helpful was to write out a cheat sheet before each exam with all the theorems and definitions and example problems that I would need to know. Even though I wasn’t able use the sheet during the exam, simply the act of writing it all out in one place helped me organize the information in my mind. Again, if you put in the time, you will succeed in this class.

• From William D. (Major: Applied Statistics, Winter 20)

Me: How to be successful in MTH 2775 (with an A)? What is your advice to prospective students in this course?

William: In order to get an A in Linear Algebra, you must put in the time to understand all theorems that are covered throughout this course. If you can understand and remember theorems, it will be much easier to do the proofs on homework or exams. Instead of focusing on memorizing the steps of a calculation, try to understand the definitions and theorems. This will help you start a problem without feeling stuck. When you are having a difficult time understanding the different concepts in this course, go to Professor Tran's office. I was a frequent visitor to his office and that is a major reason for my success in this course. Professor Tran is willing to help and will explain concepts that you do not understand from the notes/book. Also, do all homework and practice exams that Professor Tran assigns. The practice exams were my favorite resource to use while I was taking Linear Algebra. They are typically a similar difficulty or maybe even more difficult than the exams, so be sure to thoroughly understand each problem on every practice exam. If you work through the homework and practice exams until you really understand the problems, then the exams will not be intimidating.

Me: Which sections are most difficult? How do you study them?

William: Section 3.5, Change of Basis was one of the more difficult sections for me to understand. I was very confused on what everything meant in class, so I made sure to take my time while doing the homework for this section. I was also asking questions about problems from 3.5 during class and in Professor Tran's office up until exam day. Whenever you hear Professor Tran say something along the lines of "This is the key..." or a "Remark" in the notes, I recommend paying very close attention. When I would study more difficult sections, I would try to look at it in a way where I can apply a definition. Again, if you can understand the definitions that you will learn throughout this course, it will make your life easier.

Another strategy I used during this course on the difficult section was making small note cards with definitions or theorems written on them. If you use these note cards while you do your homework or practice exams, then these theorems and definitions will eventually be memorized. I highly recommend rewriting definitions or theorems in some way outside of class in order to understand them thoroughly.

• From Megan Jaworowicz (Major: Actuarial Science, Winter 20)

Me: How to be successful in MTH 2775 (with an A)? What is your advice to prospective students in this course?

Megan: The most important aspects of Linear Algebra that you need to understand and know are the theorems, definitions, and main points or remarks Professor Tran points out. If you know these, most of the problems should be relatively solvable no matter what he throws at you. Yes, they are difficult and complicated, but there are many resources that will help you. What I did to remember the important aspects was I wrote them down on notecards and also highlighted them in my notes when I reread them. I also highly suggest doing the homework before the next lecture, because you can ask any question you want about the homework at the beginning of the lecture. Also attend office hours! I have gone and seen many professors to ask questions on homework or concepts and by far has Professor Tran been the most helpful during these times. Be smart about what questions you ask though. Yes, you can ask about a specific homework question, but maybe ask about one that you have to apply multiple concepts too. Or just ask about a theorem or definition and just say, “Hey, I don’t get this at all”. That was extremely helpful to me when I didn’t understand how to find the range of a kernel.

Besides asking questions, it is important to learn the information on your own as well. Make sure you do the homework. It is extremely beneficial! The quizzes and exams are extremely similar to the homework and sometimes even an exact homework problem will appear on the exam. It is imperative to do the homework. It is also important to do the practice problems Professor Tran gives you and to get them done before the review day.

Lastly, I studied with fellow classmates. This is very beneficial, especially when doing homework. If your trying to find say the determinant of a 4 x 4 matrix and you get the wrong answer, but your friend gets the correct answer, it is an easy way to find out where you went wrong. Another way a fellow classmate is able to be helpful is you can talk to them. I usually give this advice to people I tutor, and I have always found it extremely helpful especially when its math. If I don’t understand what I’m doing wrong on a specific homework problem I talk it out with someone. This person your explaining it to doesn’t even have to know a single thing about math, they just have to listen, but in this case, you can use your classmate who knows what you’re talking about. This helps you figure out where you went wrong in your work and find your mistake. How this is also helpful is you can have a conversation with your classmate about the theorems, definitions, and main points in Linear Algebra. Just talk through them and you will realize that you and your friend thought about a theorem completely differently but were still coming to the same conclusions in the homework. This helps you learn the material better and also helps you remember it too.

Me: Which sections are most difficult? How do you study them?

Megan: For me, the class was the most difficult at the end, especially the proofs. I struggled with them the most. The proofs that dealt with Section 5.5 (Orthonormal Sets) and Section 6.3 (Diagonalization) were the hardest for me. For these, I resorted to the textbook. I read both of the sections in the book which helped me to get a broad understanding of how to approach the proofs and what needed to be proved, but I was still getting stuck on how to prove it. I would have a starting idea, but then I would get stuck. Then after a while, I would get together with a classmate and we would talk it out. Seeing the different ways we both approached the proofs helped me get better at doing proofs and working my way through them.

This class just takes a lot of work and practice, but anyone has the potential to do well in it. Good luck and have fun!

Me: What is your advice to prospective students in this course?

The best way to be successful in this class is to put a lot of time and effort into learning, as well as make sure your notes, homework, etc., are organized so that it is easier to learn from. Unlike other math classes where memorization can be used, linear algebra has a lot of proofs and many varied types of questions. Because of this, it is better to remember theories and really understand how the concepts work so you can apply it to all the different types of problems. The assigned homework for this class is extremely helpful and in order to be successful, you will not only have to do all the homework but also make sure you fully understand each problem. If you struggle with a problem, it is best to talk to Professor Tran during office hours or ask the question at the beginning of class.

For tests, it is also important that you do the entire study guide and again ask Professor Tran questions on review day. The best way to make the study guide as useful as possible is to do it after you have studied as much as you can. This way, you can see if you have studied enough and do fully understand all the concepts. If there is a section on the study guide where you did not perform well, I would look over homework again and find extra problems in the book to do. I also sometimes did the Practice Exam A and B that are at the end of each chapter in the book. Another way I studied for each test was by compiling my own study guide. I would look at the book, my lecture notes, and homework and rewrite all theories, as well as problems that I found very useful. By doing this, you will have a compiled and neat source to study from. This also makes it useful to study for the final because most of the notes you will already have! Many of the concepts in class will use previous concepts, so it is very important to stay on top of things, which I found was easy to do when I summarized each section of the book.

Which sections are most difficult? How do you study them?

I struggled a lot with the concepts from Chapter 5, such as orthogonal subspaces, orthonormal sets, and inner product space. This chapter has a lot of proofs, which is usually something I struggle with. The best way to learn the concepts is to try and really understand each of the theorems, as well as do as many practice problems as possible. A lot of times, getting help from different sources is also useful. Everyone has a different way of explaining, so between Professor Tran, the book, your classmates, or online resources, you are very likely to find a way that makes the most sense to you. When I struggled in specific sections, I also often did extra problems in the book. While they are not assigned, there are often many problems that are similar to the assigned ones and it is always good to do extras until you understand the concept.

• From Tessa P. (Major: Computer Science, Winter 20)

Me: How to be successful in MTH 2775 (with an A)? What is your advice to prospective students in this course?

Tessa: The best way to be successful in this class is to put a lot of time and effort into learning, as well as make sure your notes, homework, etc., are organized so that it is easier to learn from. Unlike other math classes where memorization can be used, linear algebra has a lot of proofs and many varied types of questions. Because of this, it is better to remember theories and really understand how the concepts work so you can apply it to all the different types of problems. The assigned homework for this class is extremely helpful and in order to be successful, you will not only have to do all the homework but also make sure you fully understand each problem. If you struggle with a problem, it is best to talk to Professor Tran during office hours or ask the question at the beginning of class.

For tests, it is also important that you do the entire study guide and again ask Professor Tran questions on review day. The best way to make the study guide as useful as possible is to do it after you have studied as much as you can. This way, you can see if you have studied enough and do fully understand all the concepts. If there is a section on the study guide where you did not perform well, I would look over homework again and find extra problems in the book to do. I also sometimes did the Practice Exam A and B that are at the end of each chapter in the book. Another way I studied for each test was by compiling my own study guide. I would look at the book, my lecture notes, and homework and rewrite all theories, as well as problems that I found very useful. By doing this, you will have a compiled and neat source to study from. This also makes it useful to study for the final because most of the notes you will already have! Many of the concepts in class will use previous concepts, so it is very important to stay on top of things, which I found was easy to do when I summarized each section of the book.

Me: Which sections are most difficult? How do you study them?

Tessa: I struggled a lot with the concepts from Chapter 5, such as orthogonal subspaces, orthonormal sets, and inner product space. This chapter has a lot of proofs, which is usually something I struggle with. The best way to learn the concepts is to try and really understand each of the theorems, as well as do as many practice problems as possible. A lot of times, getting help from different sources is also useful. Everyone has a different way of explaining, so between Professor Tran, the book, your classmates, or online resources, you are very likely to find a way that makes the most sense to you. When I struggled in specific sections, I also often did extra problems in the book. While they are not assigned, there are often many problems that are similar to the assigned ones and it is always good to do extras until you understand the concept.

• From Robert Roy (Major: Mechanical Engineering, Winter 20)

Me: How to be successful in MTH 2775 (with an A)? What is your advice to prospective students in this course?

Robert: As you will hear many times from professor Tran himself, doing the homework is the best thing you can do to prepare yourself to be successful in this class. The homework problems assigned are very similar to the problems on the exam, sometimes almost exact. Of course attempt all the homework problems, but stress the ones that are graded (for our class they were in bold numbers) and make sure you understand the theory behind solving them. There is a reason why these problems are in particular graded, and it’s because the professor sees them as the most important. This is also true for the practice exams, and to study, it is worth choosing problems from the homework that are similar to the practice exams.

At some point after each class, I looked in the book and reread the important theorems, corollaries, and derivations that were covered in class. Linear Algebra is very proof heavy, and understanding the proofs or theorems are very helpful in solving the problems later on the exam. Also, doing the homework on days other than lecture really helped me cement the material and make sure I actually understood it enough to use it to solve problems. When in doubt, ask Professor Tran, he’s very willing to help after class and meeting out of class.

Like I said, this class is very proof heavy, and much less actual computation work is done than previous math courses. This means conceptually, your knowledge should be very good in comparison to carrying out computation work if you want to get an A. I’ve found that attempting to visualize what is actually happening when things are done, such as the determinant, vector spaces, and Eigen-everythings, is extremely helpful to solving problems. 3Blue1Brown has a wonderful series on the essence of linear algebra, and watching the corresponding series video to the topic covered in class helped me answer the question “what are we actually doing?” Another fantastic free resource - other than going to class which is vital - is the MIT 18.06 Linear algebra lectures with Gilbert Strang, which goes into a lot of depth conceptually for the topics we cover in class.

Me: Which sections are most difficult? How do you study them?

Robert.: 3.5 Change of basis, 3.6 Row space and Column space, 5.2 Orthogonal Subspaces, and 5.4 inner product spaces were the hardest sections to me. If you’ve taken apm 2554 with matrix algebra, you might have been able to get by with previous knowledge up until 3.5, but to me this is where the class starts to get harder. Understanding the bare bones concepts about what is visually happening helped me a lot (this is where 3blue1brown could help) and doing the homework problems really helped. The more problems I did the more I started to wrap my head around the topics. Understanding the theorems and corollaries helped a lot in my approach to these problems since you will use the theorems to solve them. Once again, if you do not understand a certain problem, strike up a conversation with professor Tran, or even another student to see if you can get a better grasp of the problem.

If you don’t understand one way of solving a problem, there is a high chance that Stanford, or another school has a different solution to the same problem and you might understand that way. Good luck, happy learning!

• From Parker Szachta (High school student, Fall 19)

Me: What is your advice to prospective students in this course?

Parker: There are multiple things that you need to do to be successful in MTH 2775. It may seem obvious, but the most important thing to do well is to know the required material inside and out. You never know how different concepts are going to be tested on assessment days, so you need to be prepared.

The best way to understand the content -- besides going to class, which is extremely important -- is to do the homework. If you’re stumbling through homework problem after homework problem by simply looking up how to solve everything, you might not understand the concepts at the core. Although, checking your solutions is still important! Often, the ideas of how to solve problems -- including both computation and proofs, both of which appear in the course -- seem confusing at first. But, there are often a few pieces of information at the core that are the tricks to unlocking the methods that eventually make sense.

It’s easy to get bogged down and confused with all the different definitions and examples, so focusing first on the basic ideas help to make sense of the more difficult material. It’s important to not just memorize the more abstract definitions, but also to understand what they truly mean. Because this course introduces some rather formal ways to think about mathematics, it’s important not to get overwhelmed. If something seems completely random, it’s likely a lot simpler than it may seem at first.

Beyond going to class and doing the homework, I also did some other methods to study. I completed the practice problems and practice Final Exam which truly tested what parts of the material I already understood and what parts I needed to review the most. When you do this, it’s easy to spend most of the time on the concepts you’re good at, because you already are comfortable doing them. However, most of your efforts should go towards the problems you need the most help with. Lastly, I made a formula and process sheet which helped with my studying before the Final Exam.

Me: Which sections are most difficult? How do you study them?

Parker: The most difficult sections were towards the middle of the course, because they applied matrices to more abstract ideas. For me, they were 3.3 (Linear Independence), 3.4 (Basis and Dimension), 4.2 (Matrix Representations of Linear Transformations), and 5.2 (Orthogonal Subspaces). Sections 3.3 and 3.4 were, in my opinion, the course’s first major difficulty spike, since it was hard at first to wrap my head around how the different conclusions were made. But, after a lot of practice, they began to make sense! Going back to the basics of what the definitions mean is crucial here. Section 4.2 was also on the harder side, since it applied matrices to an abstract idea. Lastly, Section 5.2 took a lot of practice to understand. That’s the key here: practice makes perfect!

Good luck! You can do it! Have fun, and learn a lot!

• From Julia Thompson (Major: Actuarial Science, Winter 20):

Me: How to be successful in MTH 2775 (with an A)? What is your advice to prospective students in this course?

Julia: I got an A in this class by dedicating a lot of time to making sure I understood as much as I could. It starts with doing your best to pay attention in class and highlighting things in your notes that Professor Tran says are very important. He gives lots of helpful hints. Next comes the homework. Go over the notes and have them out before starting. When you get stuck on a problem, don’t be afraid to use the internet. You have to be sure you actually understand how you get to each answer, though. Copying solutions blindly from the internet does you no good (and is considered cheating). If you really want an A in this class, before each exam you should look over/redo just about all the homework questions assigned (yes, this takes time, but you’ll be surprised how much faster the homework goes the second time around). To save time, skip problems that are repetitive if you think you have them down. Remember that anything on the homework is fair game for the exams. Something I like to do to study is take a sheet of paper and fill it while I’m redoing homework problems with helpful reminders, theorems, and things I tend to forget. The exam reviews are also very helpful. If you understand the majority of the homework assignments and the review problems before each exam, you will do well!

Me: Which sections are most difficult? How do you study them?

Julia: I think the class starts to get pretty challenging once you get to section 3.3 about linear independence. It definitely took me a while to wrap my head around those concepts. For me, it just took time and effort for it to eventually sink in. That’s what it takes for a lot of the complex ideas in this class. My advice for section 3.5 is to memorize and understand the “key” that Professor Tran will give you. Finding the range in chapter 4 also gave me some trouble, but after I went to office hours I got some clarity and it wasn’t so bad anymore. Also, the proofs in chapter 5 and 6 were pretty difficult. Proofs are all around the most difficult part of linear algebra. I’d recommend studying the definitions in your notes and making sure you understand them thoroughly before even attempting the proofs in your homework. Then, if you are still struggling, use the internet and/or go to office hours to see the solutions. Sometimes even if you fully understand the concepts, you still might struggle to solve a proof and that’s okay! Just make note of how it was solved and keep practicing! Good luck!

• From Nick Williams (Major: Actuarial Science, Winter 2020)

Me: How to be successful in MTH 2775 (with an A)? What is your advice to prospective students in this course?

Nick: To be successful in this class, you will have to be ready to put a good amount of effort into learning. In my experience, Linear Algebra is much more abstract compared to the calcs. In calculus classes, if you don’t understand what's happening you can do practice problems and then just go through the motions on tests because many of the problems are pretty similar. However for Linear Algebra, going through the motions isn’t really an option; after the first two chapters you really need to have some understanding about the topics and then be able to apply that to the problems. Many of the problems are things like “Prove that this matrix has this property given that it has some other property” so you really have to have a good understanding on how the math works since you aren’t just plugging numbers in.

In order to get a good understanding of the topics, I highly suggest showing up to lectures and taking good notes. Obviously copy down the notes as he is writing, but pay extra close attention when he says “remark” because what he says next is usually some piece of information that is more important than others or some kind of tip that will help you with the problems later. Second, do the homeworks! This class is no different than any other class in the sense that the professor assigns problems that they think are important to know how to do and similar ones could end up on exams. A little advice though is that you should try to do the homework before the next class because Professor Tran will take a minute to answer homework questions before starting the day’s notes. So if you run into trouble doing a problem, you’ll be able to ask it during the next class. And finally, utilize office hours! Towards the end of the semester when things got pretty hard, office hours were a great way to get clarification on how to do stuff. Even if you don’t have any specific questions, I recommend going before an exam and just listening to the questions other students have, it’s essentially only a 5 or so person lecture so he can go slower and explain things more carefully if needed.

Me: Which sections are most difficult? How do you study them?

Nick: For me, I found 3.4: Basis and Dimension, 3.5: Change of Basis and 3.6: Row Space and Column space to be especially difficult. In order to learn and study them I read the sections again outside of class before doing the homeworks. After that I asked classmates questions. Fortunately the people that sat by me knew what they were doing and I was able to meet with them to get help on how to do problems and just a general explanation from someone different than the professor. Sometimes it was helpful to just ask a fellow student who can explain something in a different way than the professor. After that, I would go to office hours and ask any questions I wasn’t able to get by either studying by myself or with a classmate. I found that between studying and asking different people for help, I was easily able to get the help I needed in order to do well in the class.

Again, it might be a hassle, but if you make a conscious effort to show up and pay attention in class, do homework before the following class rather than waiting until the weekend and then going to office hours when needed, you will have no problem at all in this class. Linear Algebra is a tough class much different than past math classes. Although difficult, it is by no means impossible to do well in; you will just have to put in work outside of class and make an effort to use the resources provided for the class such as office hours and practice problems/homeworks.

• From J. Zhang (High school student, Winter 2020)

Me: How to be successful in MTH 2775? What is your advice to prospective students in this course?

J.: There are two key components to success in this course, similar to many courses in general. It is necessary to both deeply understand the material as well as have efficient and effective testing strategies.

The most fundamental step toward achieving in Linear Algebra is to grasp the content really well. That means doing the homework and listening to lectures not as the entirety of your studying regiment, but rather approaching it as a starting point from which you can ask questions about the material. Don't just memorize the process for solving a particular kind of problem as you would memorize a script to strictly follow when you receive the test. Often, a few of the problems on the test will ask you to think about the concepts you know from a slightly different perspective. Developing the instincts to recognize which mathematical tools to apply to a novel problem can only be accomplished through slowly and deliberately asking questions at every concept your homework asks you to practice. So, when you're looking at your homework, don't just look at it as some kind of drudgery you have to get through and turn in. Instead, ask yourself, "Fundamentally, why does this work?" and "What if the parameters of this problem were slightly different?" Thinking deeply about the concepts in the course pays huge dividends when you're sitting for exams and for your future academic career in general.

The other bit which is necessary to success concerns test-taking strategies. If you had already done the first part, you would have already developed the bedrock understanding for general problems on the test as well as those few curveball questions, but there are certain practical, somewhat obvious, measures you can take to boost your performance on exams. Make sure you don't enter the test without some warmups to get your brain thinking about math. I've seen some people suggest that you take a break before big tests, which could be beneficial, but at least make sure that in the last 10-15 minutes, you're reviewing the processes and methods of thinking that are likely to appear on the test. For this purpose, having a buddy you can talk to and quiz each other in the minutes before the test can be incredibly helpful. During the actual test, make sure that you can zip through the easy questions with great speed. This is largely the opposite of the best way to do homework, which is slow and deliberate, but on the test, speed matters, and time is in short supply. Doing easy questions with a slightly hurried pace will allow you enough time to tackle harder problems on the test without losing too much accuracy on the easy ones. The hard problems are where dealing with diverse problem sets in your homework help the most. Often, you will be asked to prove something novel based on some basic assumptions. Allow the assumptions that the question gives you to guide which path you go down to prove it. Having that experience with your homework will allow you to quickly recognize which theorems and definitions you have available to you. After the test is over, make sure you let go of the stress built up from the test, and engage in whatever activity you find can help with that. Remember that at this point, it is impossible for you to affect your score.

Me: Which sections are most difficult? How do you study them?

J.: Many of the sections are somewhat difficult to grasp. Things like row space and column space, rank, eigenvalues and eigenvectors, and diagonal matrices are easy to conceptualize from their definitions but hard to really understand in your gut because they are very abstract and non-geometrical. One very helpful way I found to remedy this shortfall in intuition was to find some animations on the internet to build an intuitive understanding for some of these concepts. That and rigorously asking questions and thinking deeply about homework and example problems, as I wrote in the first section, are the most effective ways to study these concepts in my view.

• From J. Zhan (High school student, Winter 2020)

Me: How to be successful in MTH 2775 (with an A)? What is your advice to prospective students in this course?

J.: Linear algebra is not a calculation based class that you can get an A simply by memorizing stuff. To be successful in MTH 2775, the most important thing is to understand the concepts. If you fully understand the definitions and ideas, sometimes you don’t even need to memorize the theorems.

There are many new definitions you’ll learn in this class, and many of them are not easy to understand. Some new definitions and ideas are built on each other, so taking notes and having the notes organized helps a lot. Taking notes in class doesn’t mean simply copying down everything, make sure you’re paying attention when professor Tran is explaining them. Understanding the idea is more important than writing things down. Theorems are also very important in linear algebra. Professor Tran explains the theorems and proofs really well in class, the example problems also help to understand how to apply the theorems. I highly recommend writing down the proofs with the theorems in your notes. It is always better to have organized notes for this class.

Other than paying attention and taking notes in class, doing the homework is also helpful. Homework problems are more similar to the quizzes and tests compared to the examples in class. Don’t do the homework just to get it done. You can practice the new concepts when doing homework, and you’re able to see which part of the section is harder for you. More practice also helps you to do the problems faster when taking a test.

Me: Which sections are most difficult? How do you study them?

J.: Section 5.2, orthogonal spaces, was hard for me. It is hard to imagine or visualize what the space are like when it gets to higher dimensions. Doing homework allowed me to practice more about it, and good understanding of definitions and theorems in previous sections (3.6) helps me using the new theorem in 5.2. Make sure you understand everything from the beginning in the class, it actually makes the rest of this class easier.