Theoretical examples from lesson 1.2 (continued)

Hello all! I originally planned to make this post a short video, but after filming an hour-long explanation of lesson 1.3, I made the decision to think realistically about the aesthetic, and emotional appeal of over an hour of explanation of linear algebra! So instead….a quick article.

Last week, my meeting with Dr. Prudhom centered primarily around really building up a better understanding of the geometry of vectors. The book gave this example…

You are given a vector: [1, 3, -1], and you must find the vector that is perpendicular to it. The book generously gives us a method for this, explaining that the dot product method can help one find the perpendicular of a vector (the dot product is an equation that multiplies: x_1(v_1)+x_2(v_2)+x_3(v_3)) and so on and so forth ad finitem). To find the perpendicular to a vector you set this equation to zero, so in this case we would have: 0=x_1(1)+x_2(3)+x_3(-1). Our goal is to find what x_1, x_2, and x_3 are, because these are the components of the perpendicular vector. For the purposes of clarity, I changed x_1, x_2, and x_3 into x, y, and z.

Here is the first method that one can solve this problem with:

This is the way that I attempted to solve the problem before meeting with Dr. Prudhom. The issue with what I did was that I tried to solve for each variable in terms of the others. In other words, I solved for x in terms of y and z, y in terms of x and z, and z in terms of x and y. I then put these expressions into vector form: you’ve probably guessed my mistake: I ended up with the same number of variables as I started with! When we solve for systems of arbitrary size, the goal is always to come away having decreased the number of variables from when you started. The correct operation, is to choose ONE variable to solve for in terms of the others: I chose z. Then when I wrote the vector representation of the solution I had only two variables that I was dealing with, which was the closest that we could come to finding a solution.

The second method that Dr. Prudhom showed me unlocked something in my brain regarding geometrical representations of vectors:

First, we drew the x, y, z, space. This represents the plane along which the vector [1, 3, -1] lies. We plotted the vector: over 1 on the x, over 3 on the y, down 1 on the z. Then we thought about the possible solutions to the equation that we were working with: 0=x+3y-z. This equation takes place in a 3D space (x, y, z), as stated. But Dr. Prudhom explained one crucial law: the dimensions of the solution set of an equation are one less than the dimensions spanned by the variables in the original equation; in other words, the solution set to this equation would be two dimensional. Then we (meaning the royal we…pretty sure Dr. Prudhom already knew) brainstormed what an infinite 2D surface would look like. After taking far too long to realize that it would be a plane, we set about drawing this infinite plane on the x, y, z space. We drew a line coming out of the original vector that we had been given, and thought about how a plane would be positioned to be perpendicular to it. The drawing that you see here is that plane.

The most amazing thing that I realized was when I connected the two methods: the law that Dr. Prudhom gave me stated that the solution space had to be one less than the space spanned by the original variables: 2D. But in other words…you end up with two variables. That was exactly what we discovered in method one. The final vector that we wrote was in terms of two variables, whereas the original was three. For me, this connection unlocked an understanding of vectors that (hopefully) will serve me going forward. Anyway, just wanted to share that with you all!

Linear Algebra Proposal

A Study of Linear Algebra: 

Content Advisor: Dr. Prudhom 

Guiding Questions of the Course: 

  • What is linear algebra?
  • Learning strategies for how to struggle on my own with problems, and learn from a “college” textbook?
  • What does it look like for two people to struggle and work through problems together as mathematicians? 
  • What are applications of linear algebra?

Preliminary List of Resources: 

Main Course Text: Otto Brescher’s Linear Algebra *previously used in other independent studies with Dr. Prudhom 

Helpful resources

  • Linear Algebra, (Khan Academy): 
  • Linear Algebra Full Course for Beginners to Experts, (Geek’s Lessons): 
  • Linear Algebra-Full College Course, ( 

Monthly Plan and Assessment: 5 Units 

DecemberChapter 1; Linear Equations+ Weekly video posts + posted updates on notes and questions
JanuaryChapter 2: Linear Transformations + Weekly video posts + posted updates on notes and questions
FebruaryChapter 3: Subspaces of R^n and Their Dimensions+ Weekly video posts + posted updates on notes and questions
MarchChapter 6: Determinants + Weekly video posts + posted updates on notes and questions
April Chapter 7: Eigenvalues and Eigenvectors + Weekly video posts + posted updates on notes and questions
May Time built in for if some of these assignments or chapters require more time than expected *if time is permitting explore other chapters 
Final Assessment: Dr. Prudhom will potentially assign a final problem set as an exam, and he will evaluate the overall learning, engagement, and understanding noted from the rest of the course, as well as the posts that I made and how I used the blog.  

How much time per week: 

The amount spent on a normal class period: 

  • minimum: 5 hours a week like a traditional class: combination: free period (E), outside of school and weekend homework time
  • meeting time: Lunches on Day three 

Major Commitments: 

  • Congressional Debate: Once to twice a week practice 
  • Winter Musical: the number of days a week changes based on the week (not a lead role)

Full Schedule: 

  • Advanced Latin Lit
  • Intermediate Mid French 
  • AP Seminar 
  • Western Lit 
  • Physics M 
  • Honors Precalculus 
  • Free period 

Description and Reasoning: 

This independent study is important to me for several reasons. Firstly, my previous Algebra two math course at my old school covered a lot of Honors Precalculus material at the discretion of the teacher, but not quite enough to allow me to advance to AP calculus. I would really appreciate a way to challenge myself in math this year. My other classes are feeling quite light on homework, and I am ahead in several classes, giving me time to explore this opportunity. Linear algebra appealed to me specifically because I am very analytical, and I want to learn more about some more theoretical, analytical concepts. My homework load is feeling very light this year, and this is a way to challenge myself. 

I respond to self guided learning, and often enjoy struggling through extra material on my own, and grappling with concepts. That is one of the reasons why teaching myself out of a college text appeals to me: I can hone my ability for self-guided learning. 

I met with Dr. Prudhom several times and we created this month-by-month plan based on his experience teaching the class, and realistic expectations of a good minimum to commit to. We are following a plan based on suggestions that the textbook gave, combined with the topics that Dr. Prudhom and I think would most appeal to what I want to get out of the course. I am, of course, open to any suggestions surrounding the curriculum, as my whole goal is simply to learn and grow in this area. 

Video # 1 + 2

After some technical difficulties: here are the first videos!

Hello and Welcome!

Hello and welcome to my Independent study! My name is Taylor Winstead, and I am so excited to be studying linear algebra this semester with Dr. Prudhom as my advisor. I have a passion for teaching, and sharing my learning, so the way that my posts will work is that each week I will upload a video explaining what I have been learning; sharing practice problems, explaining my notes, and guiding you along this process with me. I will also add additional things like notes from my meetings with Dr. Prudhom, and potentially some updates on things that have been confusing me, and how I worked through them. As of now, my posting plan is for Saturday, since I will have just learned new things all throughout the week, so you can look forward to my first video tomorrow. **Preview: it will be on vectors, augmented matrices, Gauss-Jordan elimination, reduced row echelon form, and more with two practical example problems, and one theoretical one. Stay tuned!