MAT223 Linear Algebra Exam Review questions answered in videos

Dear students from MAT223:

I’ve received some requests to explain the following questions and I made a few short videos to explain how to solve them.

Happy learning!

  1. If $T$ and $S$ are linear transformations $ \mathbb R^3 \rightarrow \mathbb R^3 $, and $\displaystyle \vec{u}, \vec{v}, \vec{w}$ are three distinct vectors in $\mathbb R^3$ so that

 T(\vec{u})= S(\vec{u}), T(\vec{v})= S(\vec{v}), T(\vec{w})= S(\vec{w}),

then $\displaystyle T(\vec{x}) =S(\vec{x}) $ for all $ \displaystyle \vec{x} \in \mathbb R^3.$

2. If A is 3 \times 3 matrix and dim(null(A-I))=3, then A is diagonalizable.

3. Consider

     \begin{equation*}A=\left[ \begin{array}{rrr}2 & 4 & -1  \\0 & 7 & 2  \\0 & -2 & 3\end{array} \right]\end{equation*}

Decide whether A is diagonalizable. Justify your answer.

4. What is the matrix of a linear transformation

 T : \mathbb R^4 \rightarrow \mathbb R^4

with

    \begin{equation*} T(\vec{e_1})  =\vec{e_2} = 3\vec{e_3}; \\ T(\vec{e_2}) =2 \vec{e_1} - 3\vec{e_2} + \vec{e_3};\\ T(\vec{e_3})  =3 \vec{e_3};\\ T(\vec{e_4}) =-\vec{e_1} - 4\vec{e_4} \end{equation*}

Compute the rank of this matrix.

Are you living vicariously?

First I have to thank one of my favorite podcasts: Hidden Brain for motivating me to write about this post today. They recently published an episode titled “Close enough: the lure of living through others ” and it resonates so much with me.

Do you ever find yourself going through video after video about a certain project you plan to do? Perhaps you are looking for instructions on how to do it, or simply looking for inspirations from others that have done it. It’s almost mesmerizing when we watch experts do what they are best at. And we somehow feel we can do it as well after we watch enough videos which is really an illusion. I find the same analogy also applies to our students: they watch us solve problems in class, and they may even find YouTube videos on the same topic and watch a few of those. And they tend to believe they can also solve similar problems after spending so much time watching. We all know how that turns out when we mark students’ test papers. They don’t know how, even though they’ve spent a lot of time watching others do it. Watching is not equal to doing. It’s a simple fact and yet many fall into the false belief that if we watch enough, we’ll become that expert in the videos.

I have to admit sometimes I make the same mistake: when I’m attending online courses, I watch others having discussions and feel I’m also part of them, even though I didn’t post a word; I feel I’m expert in the subject matter after browsing through what’s offered in the course, without actually spending much time on the listed learning objectives, only to find myself at loss when I come across the same problem somewhere else. In order to avoid this from happening, I tend to register way less courses nowadays, so I will have enough time to really sit down and study.

Reflection of using Geo-Gebra app in Linear Algebra course

Last semester I had the chance to explore how would using Geo-Gebra in my Linear Algebra course affect student’s learning experience. My initial goal was to find out whether using it would improve student’s engagement in the class. I had taught this course for a few times by then, and one observation I made was how quiet the lectures were, compared with my other sections. In fact many students who took Linear Algebra last semester with me also took my other math course: Intro to Math Proof and we were discussing this interesting finding. They agreed that our Linear Algebra lectures were too quiet.  They told me that Linear Algebra classes were not as engaging, and interesting as the other course even though my teaching style didn’t change. I had to do something.

I managed to secure a small funding from my university and did the following experiments. Once every two weeks, I will have four TAs going in to my lecture and sit among students. We will do one Geo-Gebra activity that helps students visualize and understand a new concept. For example, when we were learning the topic of eigen-value and eigen-vectors, students were asked to go here: Exploring Eigenvectors and Eigenvalues Visually and follow the steps:

  1. Set the matrix M to be (1 0 ; 0 2)
  2. Drag the point u until you see the vector u and Mu are on the same line. Record the value of lambda. How many times do you see u and Mu lying on the same line when u travel through the whole circle? Why?
  3. Based on your observation, what can we say about the eigenvalue and eigenvector of M?
  4. Set the matrix M to be (3 5; 1 -1) and repeat what you did above.
  5. Check your lecture notes about the eigenvalues and eigenvectors of this matrix. Are the results consistent with what you observe?

TAs will be walking around the lecture hall and answer any questions students have. Some are about the applet itself, some are about the mathematics involved. By the end of the activity, a majority of students feel more comfortable about the two new concepts.

We did similar activities for a few other topics, and in general they helped students in understanding the abstract math ideas better. However, the interactions between students were not improved much. Most of them were working alone, and the class for most part was still pretty quiet. I’m in the middle of getting the data in place and measure whether the engagement level has improved or not, but my guess is there probably isn’t a significance improvement. Next semester if I’m to do these activities again, I will add at least one step: share your work with your neighbor and exchange what you have found with each other. And I will invite volunteers to talk about the questions, instead of me explaining them.

I will come back to this post once there are new findings.

My Acrylic Paintings

Credit goes to the wonderful Angela Anderson . I have been following her amazing YouTube channel and weekly lessons and learned how to paint since two years ago. She’s a role model for me on my OER journey. She’s so generous of sharing her talent in painting and teaching and has reached such a wide community. Below are some of my recent paintings.

2019 Winter MAT102: Intro to Math Proof

The main topics that we cover in this course are:

  1. Numbers, and Inequalities.
  2. Sets, functions and fields.
  3. Informal logic.
  4. Mathematical Induction.
  5. Bijections and cardinality.
  6. Integers and divisibility.
  7. Relations.

The course note is available: MAT102

The slides for weekly lectures can be found here:

  1. MAT102_Week1-2
  2. MAT102_Week2-3
  3. MAT102_Week 4-5
  4. MAT102_Week 6
  5. MAT102_Week 7
  6. MAT102_Week 8
  7. MAT102_Week 9
  8. MAT102_Week10
  9. MAT102_Week 11-12

Below are a few short video clips I made to explain certain questions upon students’ request.

  1. 2.5.29 Part a) Prove or disprove if two functions f and g are bounded, then their sum f+g is bounded.
  2. 2.5.29 Part b). Prove of disprove if two functions f and g are bounded, then f^2-g^2 is bounded.
  3. 2.5.50. Show that in any field F, the equation x^2 = 1 can have at most two solutions.
    Can you think of a field in which the equation x^2 = 1 has exactly one solution?

CV, Teaching dossier and more

Here’s a copy of my recent CV (edited in July 2021)

Here’s a copy of my teaching dossier (edited in Jun 2020)

Here are a few gems I collected during my journey as a math educator: letter1, letter2, letter3, letter4, letter5, letter6, letter7, letter8, letter9, letter10

Research has shown student teaching evaluations are biased. Teaching evaluations evaluate gender, race, and attractiveness. Men are favored. White people are favored. To see a collection of resources that discuss the issues associated with teaching evaluations, see Rebecca J. Kreitzer’s post: EVIDENCE OF BIAS IN STANDARD EVALUATIONS OF TEACHING

If you are still interested in reading the full version of my recent teaching evaluations, they are here:

MATH1500 Calculus I (2021 Spring)

MATH2150 Multivariable Calculus (2021 Spring)

MATH2720 Multivariable Calculus (2021 Spring)

MATH1510 Applied Calculus I (2020 Fall)

MATH2720 Multivariable Calculus (2020 Fall)

MATH2030 Combinatorics (2020 Fall)

MAT202 Discrete Mathematics (2020 Spring)

MAT135 Calculus I (2020 Spring)

MAT135 Calculus

MAT102 Intro to Math Proof

MAT102 Intro to Math Proof

MAT223 Linear Algebra

Are you a Repeat Learner?

Disclaimer: Repeat Learner here refers to someone who can’t stop learning new things. I don’t believe the definition: a Repeat Learner is a student who has outstanding modules from previous years gives it justice. I’m a proud Repeat Learner who repeats the activity of learning new things all the time.

My life as a student lasted long. When I looked back, I spent almost twenty-two whole years (Y6-Y28) as a full-time student. At some point when I was near the end of my student days, I thought the learning part of my life was about to be done. How wrong was I! Once I started teaching as a full-time mathematics lecturer at Singapore Polytechnic, I quickly realized there were so many things I needed to learn: how to write a lesson plan that makes sense; how to communicate with students; how to write on the whiteboard/blackboard which minimizes the chance of anything getting erased during an one-hour lecture, how to navigate the LMS (we were using Blackboard back then), etc. I had a great officemate when I started my job, and she taught me new skills everyday in the first few months. I didn’t even know how to order textbooks! Once I settled on my new role as a lecturer and knew what I was doing, I found myself learning how to use Camtasia to make video lessons; taking online courses to learn about the newest edtech tools,  and even a cool visualization software for statistics: Tableau. All these learning experiences keep my day exciting. They have brought much frustration and struggle, but also joy, satisfaction and fun. I’m in love with learning new things! It helps me master skills that make me a better teacher.

What I didn’t realize back then is learning new things can also help me stay humble and connected to my students. Sometimes I found myself quietly complaining things in my head while teaching: How can you not know this? How can you forget something that we just learned last week? How can you not get it? You see, I forget what it’s like to be a student, to be a learner who struggles. I took up painting two years ago, and whenever I can’t get things right, which happens to each one of my paintings, I tell myself this is what it’s like to be learning new things. Those quiet complaints in my head gradually go away. I’m able to put myself in my students’ shoes and see things in a different angle now. I’m more empathetic because I also struggle when I learn new things and I know that’s the good thing: without making mistakes and struggling, progress and growth won’t happen.   I’m not suggesting every teacher to go out and learn something new today, but it’s important to remind ourselves what it’s like to be a beginner, a learner.

Now I’m challenging myself to learn how to play piano, which I figured might take years, especially after my first lesson. I’m not giving up just yet. The learning part is too good to walk away from. I guess I’ll never quit being a student.

Should we all become edtech gurus?

I’m pleasantly surprised by how well-organized eCampus Ontario extend mOOC is, and hoping to make some meaningful and long-lasting connections with the community here.

As I’m going through the materials of module 2: Technologists, I started thinking of why digital literacy matters, and how much does it matter to “good teaching”.

Being digital literate starts with knowing the problems we want to solve. Is it about improving student’s engagement? Is it about deepening understanding of key concepts? Is it about communicating more effectively? Or is it about having a fun and open learning environment? While I was teaching in Singapore, every faculty member from my department took part in the popular Coursera online course: Powerful Tools for Teaching and Learning: Web 2.0 Tools.

That was the first time I took some time thinking about what exactly are digital literacies and why they are important. Our students today are different from students decades ago; in fact they are probably more digital literate than us in many aspects. In order for us to improve our teaching and reaching our students, being aware of what tools are available becomes very important.  On the other hand most edtech tools have their own limitations. Before we use any new tool, it’s a good practice to try to understand what they are meant to help with, and what potential issues we may run into. 

I believe most of the problems we are trying to solve have low-tech or no-tech solutions. Using digital tools is just one way of solving it. Perhaps before we dig in the endless list of “cool” tools that are out there, we should ask ourselves can we come up with a no-tech solution to address the issue at hand, focusing on the students, and what’s best for them.

I find the most useful way to adopt new tools effectively is by discussing it with colleagues and the bigger teaching community (we have a wonderful teacher community here: teacherforlearning channel! ). It comes to each one of us to share our experiences and spread it out, especially what don’t work. Don’t hesitate to share something that you tried and didn’t work. It might save someone else plenty of time and frustration in future. I stopped using Mentimeter a while ago because I didn’t like the paid version, and it’s hard to edit mathematics there. 

For those who are taking the same course with me, you can find my extended activities of Module 1 here:

https://docs.google.com/document/d/1zqokoWhrpin3XtxNA6S3L0BvPbQccsAEQMo6zgJ0zIQ/edit?usp=sharing

I joined the math department at University of Manitoba in 2020 Fall. Before that I taught math at University of Toronto Mississauga part-time since 2016 when my family moved to Canada from Singapore. I also taught part-time at two local colleges: Seneca College and Humber College while working at UTM. I taught math full-time at Singapore Polytechnic from 2012 to 2016. This is the space where I explore and share my journey of teaching  mathematics, conducting education research projects, and learning about OER.