I made it to the final video in the Essence of Linear Algebra playlist from 3B1B this week but only made it halfway through that final video. My entire week was essentially spent working on understanding the topic of the 12th video which had to do with a concept known as Cramer’s Rule. The sad part is after spending three or four days working on it, I still don’t understand what’s going on with Cramer’s Rule, but I have a decent grasp on how it works (i.e. how to work through the C.R. formula). I wish I would have spent more time studying this week (I only studied for ~5-6 hours), but I’m happy with what I was able to learn. As I try to explain below, Cramer’s Rule is all about finding the determinant of matrices so I got a ton of practice with that this week. I’m disappointed that it’s taking me so long to get through this playlist which is why I’m annoyed at myself that I didn’t spend more time studying this week. 😒
Like I said, I don’t really know what’s going on with Cramer’s Rule, but my general understanding is that it’s a way to solve for the coordinates of a “mystery vector” given 1) a matrix transformation and 2) an output vector. In layman’s terms, if you apply a matrix transformation to a grid it’ll result in the vectors on that grid moving to certain coordinates. If you want to go in reverse and find out what the coordinates were for a “mystery vector” BEFORE the transformation was done, Cramer’s Rule is a way to do that.
To understand C.R., you first have to remember the formula for the area of a parallelogram which is base * height. If either the base or the height is equal to 1 (a.k.a. the base was î or the height was ĵ), then the area is simply equal to the value of the opposite side. (For example, if the height in a parallelogram was 1 and the base was 5 then regardless of the shape of the parallelogram, the area would be equal to 5.) Here are two photos that somewhat address what I’m talking about:
This concept works for what’s called a parallelopiped which is what you might think of as a 3D parallelogram, where the volume is equal to width * height * length. Here’s an image that represents the value of the z-coordinate given î and ĵ:
I can’t articulate how it works, but here’s an image of how the formula for C.R. works:
Although I can’t put into words why C.R. works, I was still able to use its formula. I worked through a question that was left at the end of the video where Grant asked the viewer to find the x, y and z coordinates using Cramer’s Rule. Here’s the question with my notes and then two screen shots at the end where I used a C.R. calculator that I found online to confirm my answers:
So ya… I don’t understand how this works and I’m pretty frustrated that I can’t explain it. As you can see, I got a lot of practice using the formula to find the determinant of 3×3 matrices which was good, and I think I’m not far off from understanding C.R., but it’s definitely frustrating that after spending so much time on it I still don’t really understand Cramer’s Rule.
I learned about two other concepts this week from the other three videos I watched. The first had to do with the idea that although our conventional way of thinking about the basis vectors î, ĵ and k̂ are that their x-, y-, and z-coordinates are equal to 1 in their respective directions, that is an arbitrary convention we’ve come up with and that you could consider any set of x-, y- and z-coordinates to be the basis vectors. The second concept I learned about had to do with what are known as Eigenvectors which I’m not going to attempt to explain, although for the most part I was able to follow along with what Grant was talking about in that video. Here’s a screen shot from the video on Eigenvectors which hopefully by next week I’ll be able to explain:
Considering I haven’t really explained anything, this is one of my most disappointing posts ever. I did get a lot of practice working with vectors this week but I feel like an idiot right now and can’t at all put into words anything that I learned… AGH!!! 😔Hopefully by next week some of what I’m learning will sink in a bit more and I’ll be able to better articulate some of this. It goes without saying, but obviously I didn’t make any progress through Multivariable Calculus (100/4,800 M.P.) this week. I’m praying that I’ll finally get through the videos in this playlist by next week and will be able to get back to it. I started this playlist just so I could understand dot products so I’ve definitely gone off on a tangent from the purpose of why I came to this playlist in the first place. Even though I don’t have a great understanding of linear algebra (or dot products for that matter), I’d like to get back to M.C. and just power through the pre-M.C. section since I’ll probably start to understand it better once I move forward. Plus, I feel like I’ve made no progress lately and I’m getting pretty sick of it. 🤬