I had a pretty weird week on KA. I was more busy than usual with other things and didn’t spend nearly enough time on KA from Tuesday to Friday. I tried to make up for it on Saturday and Sunday but didn’t have much success. I made it through five videos and understood something like 10% of everything that was talked about. I think I made some progress understanding notation and how to do some of the matrix/vector multiplication, but in the big picture I have no idea what’s going on, let alone why the math works the way it does. Nonetheless, I put some decent effort in today (Sunday) copying out Sal’s work through the sixth video in the Null Space and Column Space section of Vectors and Spaces, so I’m hoping that some of what I wrote out will stick in my memory and come in useful later on down the road to the big picture of what’s going on become clearer. 🤞🏼🤷🏻‍♂️

Since I didn’t understand much of anything I learned this week, this post is mostly just going to be the screen shots I took of the videos and the notes I essentially copied directly from the videos without understanding them. The second video from the section (which was the first video I watched this week) was titled Introduction to the Null Space of a Matrix. I can’t give an explanation as to what’s going on, but here are some screen shots from that video and my notes:

The second and third videos I watched this week were titled Null Space 2: Calculating the Null Space of a Matrix and Null Space 3: Relation to Linear Independence. I wasn’t able to take much away from the vids, but I did make a note that at the end of the third video Sal said:

• “If the column vectors of a matrix are Linearly Independent, then the null space of the matrix is only going to consist of the 0 vector. Or, another way of phrasing it – if the null space of a matrix only consists of the 0 vector, then the columns of that matrix are linearly independent.”

So, ya… I don’t really know what that means, but here’s a screen shot from the second of the two videos:

The fifth video I watched, Column Space of a Matrix, was definitely confusing, but it did help me understand some of the notation used in matrices, specifically when referring to the columns inside of a matrix which can be thought of as “column vectors”. Here’s a note I took about it:

Other than that, once again, I didn’t take much away from this video. Here are two screen shots from it though:

I made a note that said, “I don’t understand [the Column Space of a Matrix video]. I think it’s saying if you have a vector, a, with ‘n’ rows, and a matrix A = mXn, and you multiply A by another vector, x₁, where x₁ also has ‘n’ rows, you can always get to a by scaling Ax₁, meaning that a is inside the span of Ax₁ = C(A).”

So, again, ya… This is a great example of how just about all of this stuff is going over my head. 😔

Finally, here are some screen shots and my notes from the final video I watched this week, Null Space and Column Basis:

Hopefully my notes do a decent job of explaining what’s going on in this one. Working through the math with Sal as he went through it was frustrating, but I do think it will have helped me retain some things that will hopefully be useful. If nothing else, the next time I go through a question similar to this one, I’m guessing I won’t be as frustrated or as intimidated since I will have done it before.

I’m hoping that I can make some better progress next week (which I’ve been saying for the last 10 years), however my job is going to be taking up a lot of my time from now until September, so I’m guess the next two weeks will be just as challenging as this week was to get through KA vids. Even still, I’m optimistic that I’ll get through it all soon enough. Although I don’t really understand the bigger picture of what’s going on with matrices, vectors, systems of equations, etc., the silver lining is that I don’t find the math behind turning matrices into RREF too difficult, or adding vectors, etc. I’m hoping that once I start to understand what is happening, the flood gates will open and I’ll quickly(ish) start to figure it all out. In any case, I’m only four videos away from finishing off this unit, so hopefully next week I’ll start my post by saying I finally made it to the next unit, Matrix Transformations! (Which sounds like it will be a pretty tough unit. 😭)