It wasn’t a great start to the New Year. 😞 I technically watched six KA videos but only made notes on five of them, and my notes on the fifth video were only half complete because I didn’t know what was going on… By the end of the week I was pretty frustrated so I Googled a few other videos about what was being talked about which helped a bit. I’ll explain below, but I’m having a hard time visualizing what a lot of the notation is representing. I try to think about what’s going on in 3D and with simple 2×3 and 3×2 matrices but I still find it hard to understand. Funny enough, I’m surprisingly optimistic that it’s all going click into place for me soon. I’ve been thinking back to the unit circle and how confusing I found that and then all of a sudden it all clicked into place for me. Working through these matrix operations and trying to understand the notation feels reminiscent of that. But to sum up, this week was not how I was hoping to start the New Year but I did manage to make a bit of progress and I feel like I’m not too far from having it all become much more clear. (Hopefully. 🤞🏼)

Video 1 – Orthogonal Complements

This was the first video in the unit, and it introduced orthogonal complements. I think my written notes above do a decent job of explaining the gist of what the orthogonal complement to a subspace is so I won’t go into further detail. In the first part of the video, Sal worked through the proof that the orthogonal complement, V^⊥, is in fact a subspace. The rest of the video was confusing to me. As far as I understand it, Sal was just explaining how the transpose of a matrix relates to the column space, null space, and row space of the OG matrix. Like I said in the intro, I have a hard time visualizing subspaces in general and these three subspaces in particular, not to mention how they relate to each other, AND how they relate to the transpose… 😮‍💨

Speaking of how they relate to each other, in the last screenshot above, you can see that Sal showed how they all equal each other in different ways which I can somewhat understand but don’t have a solid intuitive grasp on. This is the notation I’m still trying to have sink in.

Video 2 – dim(v) + dim(orthogonal complement of v) = n

I’ll let my written notes speak for themselves on this one.

Video 3 – Representing Vectors in Rⁿ Using Subspace Members

I found this video very hard to follow. In my notes, I wrote:

“I think the point of the third video is that V and V^⊥ don’t overlap. I think Sal is proving this by taking the vectors in V, denoting them with v_x, and taking the vectors in V^⊥, w_x, and then… something…

• V + V^⊥ = n in Rⁿ
• V + V^⊥ overlap ONLY at 0
• Vectors in V + V^⊥ are linearly independent (maybe?)
• Using the basis vectors in V and V^⊥  to reach any other vector in Rⁿ, there’s only one unique combination of v’s and w’s in V + V^⊥ …maybe?”

So… ya. Whatever that means.

Video 4 – Orthogonal Complement of the Orthogonal Complement

As far As I understand it, the question Sal was trying to answer in this video is, if you have any vectors in V that dot with the vectors V^⊥ to equal 0, are there any vectors outside of V in something like (V^⊥)^⊥ that you could dot with vectors in V^⊥ to also equal 0? Come tot think of it, I think the video was simply asking if there IS such a thing as (V^⊥)^⊥ with vectors outside of V. Assuming that is what the video was addressing, apparently the answer is no. As you can see from my written notes, the takeaway seems to be that (V^⊥)^⊥ and V are the same subspace.

As a side note, I find Sal’s diagram pretty confusing. You can see why based on my note below. There’s a good chance I may be misunderstanding the entire thing, however:

Video 5 – Orthogonal Complement of the Null Space

This video was similar to the other four videos and was just explaining how the null space, row space, and column space of a matrix all relate to the transpose of the matrix. I thought I had it all figured out, but as I was writing my note above, I got confused and that’s when I decided to branch out and watch some other videos. So, my note above is only about ¾‘s finished and I don’t completely understand it. (The part I got tripped up on was where it says “left null space” in the screenshot.)

I’m not going to explain them, but here are the four other videos I watched this week:

And that’s going to do it for this week’s post. Pretty abysmal but like I said int he intro, my gut tells me I’m not far off from having a lot of this click into place. I think I just need to wrap my head around the differences between the four subspaces I talked about above (including the left null space) and once I have that figured out, I’m guessing it will all seem much more manageable.

I’m not going to lie, I was pretty demoralized this week, but as I was watching the last video I posted above, I was looking at the math the guy was writing down and it dawned on me that I would literally have had 0% understanding of what he was talking about a few years ago. Even though I still don’t completely understand it, I can at least somewhat follow along and understand the notation. So, even though it sometimes feels like I’m getting no where with linear algebra, I’m still happy with how far I’ve come and the progress I’ve been able to make. But still, I’m praying I can start to make sense of it all sooner rather than later so I can finally move on. 🙏🏼 😩