I’m sad to say that I didn’t get much done on KhanAcademy this week. I made it through six videos, only writing notes for three of them. the good news is that I wasn’t stumped by anything from the videos and still learned a handful of new things, but that also makes me think that I probably should have gotten through more than I did. Not to make excuses, but I was super busy at work this week, and I’m also grinding away on this other project I mentioned in a few of my other posts. (I’m still not ready to go into detail about that project, but I will soon!) In the big math picture, I’m certain that I’m making headway understanding linear algebra and am happy that I’m not getting my butt kicked as bad as was before I watched Dr. TB’s linear algebra playlist. Nonetheless, I’m still really hoping that I’ll start to get through this course faster, especially once the holidays are over.

Here are the six videos I watched this week:

Video 1 – Expressing a Projection Onto a Line as a Matrix Vector Product

(I worked through this video on Tuesday and now can’t really remember what I was talking about in my notes. I believe the question the video was solving was, “if you have some vector, x, and you want to project it onto a line, how would you figure out the matrix for the projection?” Below each screen shot is what I had written in my notes to explain each screenshot.)

“The first two screenshots are the algebra to setup the question.”

“This third screenshot shows the math to find the matrix.”

“This screen shot is a geometric representation of wtf is going on.”

Video 2 – Compositions of Linear Transformations 1

This was the first video in a new lesson section titled Transformations and Matrix Multiplication. The five videos in this section mostly talked about transformation notation and gave a general explanation of how and what’s going on when multiply a vector by multiple matrices.

You may be able to tell that in this video Sal was simply explaining how a vector can be multiplied by more than one matrix, as long as they have the appropriate dimensions. He then worked through the algebraic proof of the two axioms of LA; transformations can be applied to vectors individually and then summed or be applied to each other and then applied to the vector, and that you can scale a vector before or after the transformation. 

Video 3 – Compositions of Linear Transformations 2

I found this video and the following ones somewhat annoying. I may be confused, but Sal seems to be saying something like, “there’s one transformation, call it T, and another transformation, call it S, that can be denoted with T = B and S = A.” If I’m correct about this, which I definitely may not be, it seems redundant to have T and S in the first place since he just switched them to B and A anyways… There could be some nuance of doing it this way that’s important for some reason but if there is, I don’t understand it.

Video 4 – Matrix Product Examples

In this video, Sal worked through an example of what he was explaining in the previous one, how to multiply two matrices together. I worked through the matrix multiplication differently than Sal did, however. He took the dot product between the row vectors of A with the column vectors of B, whereas I wrote out all the column vectors of AB (which I said equalled C) as the elements being multiplied by each other and the sums of all those products for each element of AB. (That’s probably the most unclear, confusing sentence I’ve written in the 274 weeks I’ve been studying KA, and likely inaccurate, but I really don’t know how to put matrix multiplication into words. ☹️) Working through the question in this video was really helpful for me to solidify how to do matrix multiplication.

Video 5 – Matrix Product Associativity

This was another video I found annoying. I’m pretty sure it was just a SUPER complicated way of saying you can multiply three matrices the same way you’d multiply two, and that it doesn’t matter if you first multiply AB in (AB)C or if you multiply BC first in A(BC). This seemed like a given to me and so Sal’s explanation seemed superfluous. However, one thing to note is that a lot of times you can’t switch the order of ABC to, say, CAB because the matrices dimensions have to line up as something like (m x n)(n x l), for example.

Video 6 – Distributive Property of Matrix Products

Again, this seemed like another annoying video where Sal’s point seemed pretty self-evident. I believe Sal was saying that if you’re multiplying a matrix by the sum of two other matrices, A(B + C), assuming they have the appropriate dimensions, you can distribute A to B and C (i.e. A(B + C) = AB + AC) and then sum AB with AC and you’ll get the same answer. This seems like pretty basic, straightforward algebra which is why it’s definitely possible that I’m missing the point of something here… It could also just be that Sal wanted to work through this proof simply for the sake filling in all the blanks of linear algebra and being as clear as possible. Or, like I said, something may have gone completely over my head…

And that was all I got through this past week. I’m feeling pessimistic that I’ll get through much more next week as I’m likely going to be just as busy. I’ll probably continue to be just as busy up until the holidays, as well. 😔 

I want SO badly to finish the MATH section of KA but I’m finding it super difficult to get through these final few courses. I vividly remember telling my mom five years ago that if I could learn calculus then I could do anything. I REALLY want the new project I’m working on to succeed but I’m sure that it’s going to be very difficult to make it happen. The superstitious side of me feels like I HAVE to get through KA for this other project to work, so I really want to get through it sooner rather than later. Plus, after 274 weeks of studying math, I’m ready to move on to bigger and better things! (But even when I finish the MATH section, I’ll probably just get started on physics. 🤓 I may devote a little less time to it each week though. 😮‍💨)