I once again got my butt kicked this week. 😔 I had to go back to Unit 2 at the start of the week to rewatch two videos to try and get a handle on what’s going on. I watched them each twice, spoiler alert: it didn’t work. After that, I only made it through five videos and pretty much had no idea what the hell was going on the entire time. I watched each video at least twice if not three times and still couldn’t get much of a grasp of what Sal was talking about. So, overall it was a pretty frustrating week. I’ve been ending my last few posts saying that I feel like I’m on the brink of having a linear algebra lightbulb go off for me and have it all click into place. I could be way off about this, but even after this dismal week, I still feel that way, which is hard to believe given the little progress I made… Maybe I’m further away from understanding LA than it seems, BUT I’m going to try to stay optimistic and pray to the math gods that it starts to make sense soon. 🙏🏼😩

Video 1 – Orthogonal Complement of the Null Space

This was the video I ended on last week but didn’t write any notes on because I wasn’t sure of the notation. I think I have a better grasp on the notation now, but I’m still really struggling with understanding what the column space, row space, and null space all represent in the big picture. Here’s what I think they are:

• The null space are all the vectors that collapse down to 0 or at least down to a lower dimension after a specific matrix transformation. 
• The column space are whatever column vectors in a matrix are linearly independent from each other and the null space are the column vectors that have free variables, i.e. they don’t have ‘leading-1’s’ to start their columns/rows. 
• The row space is a transposed matrix that you then take the column space of (i.e. you find the linear independent column vectors of a transposed matrix) and that ends up being the row space.

I’m pretty sure I’m right about all of that, but I don’t intuitively understand what it means and can’t visualize how these things come together or how they manipulate vectors.

Video 2 – Unique Row Space Solution to Ax = b

I watched this video three times and couldn’t make much sense of it. I’m not going to try and explain anything about it now because I don’t have much to say. The next video was an example of what Sal was talking about here though, which help I have a bit more to say on…

Video 3 – Row Space Solution to Ax = b Example

Once again, don’t really understand what’s going on here, BUT the point (apparently) is that if you have a given vector, b, that’s a part of the given matrix’s column space then… something…

Ugh… I don’t understand. Sal’s point was that from the origin you can draw a vector to the “solution set” (whatever that is) and that vector ends up being along the row space and ends up being the shortest vector you can draw from the origin to the solution set because the solution set and the row space are orthogonal. So, ya… that’s about all I got from this vid.

Video 4 – Projections Onto Subspaces

I’ll let my written notes speak for themselves on this one. (4.1 is the first screenshot and 4.3 is the last screenshot.)

Video 5 – Visualizing a Projection Onto A Plane

This video takes what Sal covered in the fourth video, a projection of a vector onto a line, and proves that it also works for a projection of a vector onto a plane and, therefore, projections in higher dimensions as well. I think the point he’s making is that the shortest distance from a vector to a line, plane, or into higher dimensions is whatever vector is orthogonal to the line, plane, or… higher dimension (🤔) and lines up with the OG vector. In 2D and 3D think of it as the OG vector being the hypotenuse of a right triangle, the projection being the adjacent side, and the orthogonal vector being the opposite side. I know that’s the general idea of what’s going on, but I don’t understand how the math works. 😒

And that’s all I got done this week… So essentially nothing considering I didn’t really understand any of it. I also rewatched two 3B1B videos from his linear algebra playlist. I had an easier time following along with those videos than the first times I watched them, but a lot of what Grant talked about still went over my head. At this point I’m not really sure what to do. I think I’m going to keep powering through the KA videos and hope that it clicks soon, but I also think I’m going to branch out a bit this coming week and try to watch one or two videos from other creators every day to see if that helps. That strategy has always been helpful in the past so I’m hoping it will once again prove useful. Fingers crossed. 🤞🏼