I literally made zero progress on KA this week and yet it was one of the most productive, enjoyable and motivating weeks I’ve had in a LONG time. As I mentioned at the end of my post last week, this week I decided to watch the Linear Algebra playlist by Dr. Trefor Bazett on YouTube. I made it through 11 of the 82 videos and found each one enlightening. As you’d assume by starting this playlist from the very beginning, the first few videos simply gave a general idea of what’s going on with vectors and matrices, and their associated notation. A few videos in, Trefor went over how to think about and solve systems of equations using matrix row operations. I found this VERY helpful. It made a lot of what I was struggling with on KA seem elementary. The final few videos I watched this week got into what’s known as “linear combinations” which talked about the span of vectors/matrices. Again, I’d gone through these concepts in KA before but the videos I watched this week seemed much more clear and intuitive. (My guess is that because I’d already been introduced to these concepts on KA, Trefor’s videos were much easier to grasp. I think it’s likely that if I’d watched his videos first and then watched the KA vids, they would have seemed just as intuitive as Trefor’s vids did this week.) In any case, I’m pumped that I FINALLY have a better idea of what’s going on with LA and am super optimistic about moving forward.
(Longest opening paragraph ever.)
Of the 11 videos I watched, I only made “note” notes on two of them (as in decent, written notes). I jotted down a the key points I took away from the majority of the other videos, however, so here are screen shots from ten of the videos with either a mini explanation of what’s going on in them or the “note” notes I took of them:
Video 1 – What is a Solution to a Linear System? **Intro**
This was the second video in the playlist. The first two screen shots show that there can be more than one solution to a linear equation. However, in the system of equations (the last screen shot) only the solution to the linear equation in the first screen shot is also solution to the system of equations. I found this video useful in helping me wrap my head around how the variables x1, x2, and x3 are distributed through the system of equations.
Video 2 – Visualizing Solutions to Linear Systems – – 2D & 3D Cases Geometrically
This indicates how two linear equations in the same plane can have either one solution (if they cross over each other), infinite solutions (if they lay on top of each other), or zero solutions (if they’re parallel without laying on top of each other).
This shows how two planes that cross through each other create a line at that intersection which would have infinite solutions on that line where the plane’s coordinates are the same.
This shows the same thing except there are three planes where they cross through each other at a single point. You can determine that at that particular point, the three planes have the same coordinates and therefore is the solution as to where the three planes equal each other.
This shows the blue and yellow planes being parallel to each other, and therefore there are zero solutions as to where they equal each other.
Video 3 – Rewriting a Linear System using Matrix Notation
This video was short but super helpful for me to understand the names for the different parts of a matrix.
Video 4 – Using Elementary Row Operations to Solve Systems of Linear Equations
This video FINALLY helped solidify in my mind how to use matrix row operations. It must have been explained to me in the past on KA but for whatever reason it didn’t stick, and this video made it seem incredibly clear. Trefor did a good job here explaining why you can scale all the variables in a row by a coefficient other than zero, and why you can add and subtract rows from each other. After watching this video, it all seemed very obvious.
Video 5 – Using Elementary Row Operations to simplify a linear equation
This is the video where it all clicked for me. I finally started to understand WHY you’d want to turn a matrix into row echelon form and HOW it works.
(Although I understand it, it’s hard for me to explain so I’m not going to try, but the notes I took from the seventh and tenth videos below show how it works.)
Video 6 – Row Echelon Form and Reduced Row Echelon Form
this clearly just explains the difference between REF and RREF.
Video 7 – Back Substitutions with infinitely many solutions
To be honest, I understand how the math works in this question but don’t really understand what the solution means. I don’t understand what a “free column” or a “free variable” is. Even though I don’t understand what the solution represents, at least I can solve it, I guess? 🤷🏻♂️
Video 8 – What is a vector? Visualizing Vector Addition and Scaler Multiplication
Clearly this just shows the geometry of vector addition. Nonetheless, I still found this video helpful watching Trefor go through it.
Video 9 – Introducing Linear Combinations & Span
The thing that was most noteworthy in this video was the description Trefor gave of a linear combination. He helped me understand that a LC is just a bunch of terms summed together to equal a constant (I think) where each term is a unique variable that can have any coefficient on it. He also made it clear that the variables can ONLY be to the power of one. As soon as a variable’s exponent is something OTHER than one, the function will have a curve of some sort meaning you’re no longer in the world of linear algebra and have gone into the calculus world.
Video 10 – How to determine if one vector is in the span of other vectors?
This video touched on what I was working through in KA a few weeks ago, determining whether a given vector is in the span of two or more other vectors. This video was only five minutes long and was quite easy to follow. Watching this video, I could easily visualize three vectors going in different directions and why scaling each one of those vectors could result in creating the vector b. It all seems so incredibly clear to me now. Praise the lord! 🙏🏼😭
And that was “it” from this week, although I feel like “it” was quite a lot! If I continue watching this playlist and wait until I finish it before going back to KA, at this rate I likely won’t get back to KA until at least October. I feel like watching this entire playlist might be the right move, however, so I’m thinking that’s what I’ll do. Most of the videos in the playlist are only five to seven minutes in length, so I can go from video to video much quicker than I was on KA. Plus, for the past 2.5 months I’ve been on my “summer” schedule, meaning I had to go to work early in the morning. This coming week I’ll be starting my “Fall” schedule, meaning I won’t be going into work until later in the afternoon so I’ll have WAY more time to dedicate to KA. Given that, it will be difficult but my goal is to get through this playlist by the end of September. That means I’ll need to watch 70 videos in 30 days which is doable but I’m sure will be a challenge. If I can keep going at the pace I went this past week, I think I can do it! 💪🏼 😤