It was a decent week for me studying LA but not spectacular. I only made it through six videos from the Linear Algebra playlist by Dr. Trefor Bazett, and I didn’t learn anything that was all that noteworthy, BUT I still got a lot of solid practice doing and understanding matrix multiplication and had some of the notation and general concepts sink in a bit more. I can tell that I’m definitely getting more comfortable and confident with knowing how matrix multiplication works which I think is helping me make progress towards understanding why it all works. Understanding how to do it definitely makes the videos less intimidating and frustrating, as well, and much easier to follow. So, all in all, it could have been better, but it also could have been worse. 🤷🏻♂️
Here are screen shots from the seven videos I made it through this week:
Video 1
I’m not going to lie, I don’t exactly remember what Trefor was saying in this video, but it looks like he was just going through how to do matrix multiplication and how you can factor out and expand matrices and vectors when multiplying them together.
Video 2
This is a pretty simple but useful example of how to do matrix multiplication. Its videos like these that really help me understand how to do the operations, probably BECAUSE they’re so simple.
Video 3
In this video, Trefor showed the full proof for how and why Ab ⃗ + Ac ⃗ = A(b ⃗ + c ⃗) which is no different than expanding and factoring integers like 2(4) + 2(3) = 2(4 + 3).
(In the past I would bold all the math in these posts, but I realized that bold text is intended to denote vectors and matrices. So I will no longer be bolding text for non-vector/matrix math like the equation at the end of the sentence above. ⬆️)
Video 4
I don’t understand what Trefor was talking about in this video, but I’ll write down the explanation I wrote in my notes for each of the screen shots below:
This was Trefor setting up the question he was going to answer. He shows on the left side that two vectors in R2 can be scaled and added to get to another vector in R2 UNLESS those vectors are parallel. The right side is the algebraic proof.
I don’t understand this part, but points 1–3 are all saying the same thing.
I think this is saying that is you take a matrix A with “m” rows, change it to REF and the bottom row is [0, 0, 0 | 1], then that’s impossible and could never happen so it’s wrong. I think this is because it ends up being a 4X3 matrix where the bottom row has a number other than 0 in the constant column and 0 + 0 + 0 ≠ 1.
This screen shot (I think) is supposed to show that as long as there’s a leading 1 in the bottom row, then any value in the constant column in that row is valid. I.e. a 3×4 augmented matrix is fine but a 4×3 augmented matrix (with a nonzero value in the bottom row in the constant column) is not.
Video 5
I’ll let my notes do the talking for me here in terms of what Trefor was going through in the video, however I wanted to add that this is the type of video that REALLY helps me wrap my head around how the notation works which, in the long run, is what makes it SO much easier to understand why the bigger concepts work the way they do.
Video 6
In this video, Trefor talked about the difference between a homogenous system of vectors (which I think means they run across the OG) and a non-homogeneous system (which I think may be a.k.a. heterogenous and does NOT run across the OG, but don’t quote me on that).
Video 7
I definitely didn’t understand the main point of whatever Trefor was trying to explain in this video, but, as you can see from my notes, I was able to better grasp the dimensions of R in matrix multiplication and why the output of the dimension of R after the operation is equal to the first matrix’s rows. Although that seems SO obvious now, I had it stuck in my head that the dimension of the output was equal to the number of columns in the first matrix and couldn’t figure it out…
And that was it for this past week. I actually did start the 20th video in the LA series, titled Linear Dependence and Independence – Geometrically, but I only watched the intro of it, meaning that’s where I’ll be starting this coming week. My goal is to get up to, or past the 30th video in the series this week. I feel like that could be difficult, but I need to start moving if I’m ever going to 1) finish off this playlist, 2) then do the KA LA subject, 3) then the differential equations subject, so I can finally 4) go back to MC and redo the course challenge… AGH! 🥵 I’ve literally been studying math every week for the past five years – by the way, it’s now my SIXTH YEAR studying math – and STILL haven’t finished my goal of ‘learning’ calculus (by which I mean completing the calculus courses on KA). I guess it makes sense that you don’t learn calc overnight but GOL–LY, it’s taking a while. 😮💨