It was unfortunately another fairly unproductive week for me. I only made it through seven videos from Dr. Trefor Bazet’s Linear Algebra playlist. However, like last week, I think I learned a decent amount considering I only watched a handful of videos. I got a much stronger intuitive understanding of what an inversible matrix is, what’s required for a matrix to be inversible, and how to determine a matrix’s inverse. Nearing the end of this week, I also watched a few videos where Trefor talked about the determinant and worked through a few examples of how to calculate it. These videos definitely gave me more insight into how the determinant is calculated, although I didn’t really get a better idea of what it is or why it works. As I’ve said a million times, I typically first figure out how to use mathematical operations before I understand why they work, so I’m happy that I made some progress this week in that regard. That said, I’m still disappointed I didn’t get through more videos. 😒

Here are the seven videos I watched this week and the notes I took for them:

Video 37 – Finding the Inverse of a Matrix

This video was really interesting and enlightening. It helped me understand why RREF and REF are valuable. As you can see from my notes, it turned out that if you append a matrix with the identity matrix, then use EROs to turn the OG matrix INTO the identity matrix, what WAS the identity matrix then becomes the inverse matrix of the OG matrix. (Pretty sure I have that correct, but maybe not.)

Video 38 – When does a Matrix Fail to be Invertible? Also more “Big Theorem”.

(INSERT PHOTO)

This video just went through the requirements for a matrix to have an inverse. I think my note does a pretty good job explaining the requirements, so I’m not going to bother writing anything else.

Video 39 – Visualizing Invertible Transformations (Plus Why we need One-to-One).

The notation in the final screen shot above is pretty tricky to understand. (For me, anyways.) The first three screen shots were taken when Trefor was explaining that if you reduce a set of vectors from a higher dimension to a lower dimension (in the example from the screen shots, he takes the vectors from R² to R¹), there is no inverse matrix that can take the vectors back to their OG coordinates in the OG higher dimension. That, I believe, is what the notation says in the final screen shot (although it’s saying it the other way around – that a matrix transformation may have an inverse if the dimension doesn’t change after the transformation), but I find it very confusing. 

Video 40 – Invertible Matrices Correspond with Invertible Transformations **Proof**

This was another video that was super enlightening and interesting to watch. I could be wrong, but I believe I was able to understand a metaphorical similarity between invertible matrices and inverse operations, AND invertible matrix transformations and inverse functions. (I’m not 100% if I have that correct, but I really hope I do because, if I’m right, it makes the intuition of invertible matrices and invertible matrix transformations WAY easier for me to understand.)

Video 41 – Determinants – a “Quick” Computation to tell if a Matrix is Invertible

This was the first video that talked about determinants this week. I could remember the general formula for finding the inverse of a 2×2 matrix and that you had to factor in 1/(ad – bc), and that (ad – bc) is the determinant. This is shown in the top of the first screen shot. I was also able to recall how to determine the determinant of a 3×3 matrix, which is shown at the bottom of the first screen shot above.

At the top of the second screen shot, the next thing Trefor did was come up with some random (-1)^{i + j} matrix. I may have missed it, but I don’t remember him explaining where this random matrix came from, however it became clear that THIS is why the order of addition/subtraction for the determinant of a 3×3 matrix usually goes (+), (–), (+). (I say “usually” because in the next video I found out that you can calculate the determinant from any column or row, but until now I would always calculate it using the first row, where the order would then be (+), (–), (+).) The equation written below the (-1)^{i + j} matrix is the linear expression for the determinant of a 3×3 matrix that uses the first row to calculate the determinant. The matrix at the very bottom of the second screen shot an example.

Video 42 – Determinants can be Computed Along any Row or Column – Choose the Easiest!

As you can see, this video continued on from the previous one. Trefor explained here that you can actually calculate the determinant of a matrix using any row or column. He worked through the same example as in the previous video, but calculated the determinant by using the second column instead of the first row. As you can see, it turns out that the value of the determinant was the same as the previous video and, as you’d guess, will always be the same value no matter what row or column you use. 

The photo from my notes shows how to calculate a 3×3 determinant using the second column. You can see that the sign of the terms goes (–), (+), (–) which relates back to the (-1)^{i + j} matrix.

Video 43 – Vector Spaces | Definition & Examples

In this final video, Trefor listed the rules for matrix multiplication and then stated that these rules also apply to polynomials. Other than that it’s interesting, I’m not sure what the point of him talking about this was. My guess as to why it’s important that the operations can be used for both linear algebra and with polynomials is because maybe that means you can place polynomials inside of matrices and the matrix operation rules will work on the polynomials inside the matrix. (Once again, I’m not sure if any of that made sense or is even slightly accurate. ☺️)

And that’s where my week ended. Like I said at the beginning, clearly I didn’t get much done, but I do think I learned a lot in those few videos. My progress through this playlist seems SO slow. I’m only halfway through the playlist which I started five weeks ago. If I keep this pace, I won’t get through the playlist until November… Considering I was hoping to get through ALL of KA by 2025, I’m pretty disappointed with how long this is taking me. I’ve also been saying the exact same thing for last few posts and I feel like a broken record, like I’m never going to finish this. To be fair to myself, I’ve been SUPER busy with a lot of other things these days. Plus, I’m still battling health issues that I mentioned in a few previous posts, so I have some sympathy for myself. But nonetheless, I’m getting tired of saying the same thing at the end of each post. I really want to achieve my goal of getting through the MATH section on KA. Fingers crossed I’ll actually make some decent headway this week. 🤞🏼