I feel like I actually did a decent job working through KA this week. I made it through seven videos, and didn’t completely understand everything BUT had a decent grasp on most of what Sal was talking about, which hasn’t happened for a while and was definitely a relief. 😮💨 I can tell that I’m FINALLY getting more comfortable with the notation used in linear algebra and also at visualizing what’s going on with the different matrix transformations. That said, I also know that I still have a long way to go before I really understand it all. So even though it wasn’t the most productive weeks I’ve had working on KA, it was nice to feel like I could actually keep up with what Sal was talking about and not have to rewatch each video multiple times and then still not know what he was talking about. I suppose the glass-half-empty way of looking at this week would be that I still don’t really know what’s going on but the glass-half-full way would be that I think I’m finally over the hump of understanding the basics of linear algebra. 🙃
(I have less than 30 minutes to write this post so I’m not going to explain the videos that I have hand written notes for. 😬)
Video 1 – A Projection Onto a Subspace is a Linear Transformation
Video 2 – Subspace Projection Matrix Example
Video 3 – Another Example of a Projection Matrix
Video 4 – Projection is Closest Vector Subspace
What I wrote out in my notes:
“The point of the fourth video (6th in the playlist) is that a projection of a vector onto a subspace, say x in R3onto a plane in R3, is the closest vector in that subspace to the OG vector x.”
Video 5 – Least Squares Approximation
Again, what I wrote in my notes was:
“[This video] starts out by saying if you’re trying to find a vector in a subspace that’s closest to a vector out of a subspace (i.e. what the previous video talked about), you do so by finding the projection of the vector out of the subspace onto the subspace. This is denoted with:
- Ax* = b
- x* = “x-star” = closest approximation to b
- (a.k.a. the “Least Squares Approximation”)
The bottom line of [this video] is if Ax* = b has no solution (i.e. b is out of the subspace), the least squares solution can be found by multiplying both sides of the equation by AT:
- ATAx* = ATb
I got lost on the algebra [in this video] but was able to understand ~80-90% of it, and I have a pretty decent grasp on what Sal is talking about. (The vector coming out os a plane helps me to visualize what’s going on – the projection, and the sin, adj, and opp sides [of the vector].”
Video 6 – Least Squares Examples
(In my notes I wrote, “Sal sets up this question by graphing the lines (2x – y = 2), (x + 2y = 1), and (x + y = 4) and showing that they don’t intersect, a.k.a. they have ‘no solution’. He then solves by finding the least squares solution.”)
Video 7 – Another Least Squares Example
And that’s what I got done and where I’m going to end it for this week. Like I said, I could have done better but I’m also not too disappointed with how it went overall. Getting through those seven videos also got me through that section of Unit 3, meaning I’m now into the third section of the unit, Change of Basis. I only have 23 videos left in the entire unit which is also the LAST unit of LA! I’m getting so close to the end but when I finish, I may spend a bit of time reviewing other LA vids on YouTube to hopefully get a stronger grasp on everything. As pumped as I’ll be to get through the LA on KA, I’m going to pretty bummed out if I still don’t understand it very well… In any case, I’m hoping I can make some decent progress this coming week both through the videos and in my actually understanding of what’s going on. I’m pretty optimistic that I will but even still, as always, fingers crossed! 🤞🏼