I feel like I didn’t get much done this week but did get a solid grasp on what’s known as ‘the determinant’ in linear algebra. Although I’m bummed I didn’t learn about much else, I think the determinant of matrices will be really important to understand going forward so I’m happy to now better understand it. My week started out watching the nineth video in the 3B1B Essence of Linear Algebra playlist which was about dot products. In it, Grant said that in order to understand dot products you need to understand what the determinant of a linear transformation is. He suggested reviewing the sixth video in the series which explained how to calculate the determinant so the majority of my week was spent reviewing that video. I’m still having a hard time wrapping my head around why matrices work the way they do, so I still don’t really understand why the determinant works the way it does. But, like always, the first step for me is always understanding how a concept works before I intuitively understand it so I feel like I’m on the right track.

Here are some notes I took that explain the features of the determinant:

As far as I understand it, to think through the determinant of a 2×2 matrix you first have to picture a parallelogram using the two vectors that makeup the matrix, as you can see from the screen shot above. The determinant is simply the total area created by the parallelogram. You can also reduce the area of the parallelogram if the matrix contains values <1. One of the harder concepts I find to understand is that if the determinant is negative then the parallelogram gets ‘flipped’, so to speak. You can think of this as a piece of paper first being flipping around so that what was the back of the sheet of paper when looking at it becomes the front.

Here is the formula for calculating the determinant:

As you can see, the formula for calculating a 2×2 matrix is very straightforward. The formula to calculate the determinant of a 3×3 matrix (which I talk about below) is much more complicated.

Understanding how to calculate the determinant is easy to understand but understanding why it works is much harder. Here’s a screen shot from the same video that breaks down why it works and a few pages of my notes where I tried to work through some examples on my own:

Working through these examples helped me appreciate why it works, but I’m still having a hard time visualizing the operation. To use an example I’ve used a million times, I can visualize the unit circle and why sin(5π/6) = –½ but I find it hard to quickly picture how the determinant formula scales î and ĵ.

I mentioned that there’s a way to calculate the determinant for a 3×3 vector. As far as I understand it, the general concept of scaling is the same when it comes to the determinant of a 3×3 matrix, it’s just used for 3D objects instead of 2D. Below is a screen shot that shows the formula to find the determinant of a 3×3 matrix. I didn’t practice using this formula so I haven’t memorized it in the slightest. And as you can see, there’s a lot more to this formula than to the formula used to calculate the determinant of a 2×2 matrix:

After rewatching the video on the determinant of linear transformations, I eventually got back to the video on dot products however I still wasn’t really able to understand them. The video I watched talked about a concept called ‘duality’ which I think means something like a 2×1 vector (i.e. two numbers written on top of each other with the top number representing the x-coordinate and the bottom number representing the y-coordinate) can also be written as a 1×2 vector (ex. [a, b]), and that if you multiply either a 2×1 or a 1×2 vector by another vector (which I believe it taking the dot product) the sum of the matrix multiplication will be the same regardless of whether you use a 1×2 or 2×1 matrix.

What I just wrote was really confusing and I’m guessing it wasn’t accurate, but it’s something like that! I’m not going to pretend like I know what’s going on, but here’s a photo from my notes where I copied out what Grant was doing in the dot product video. In the video Grant used variables but I tried to put some actual values into the example to concretize what was going on:

I’m hoping that by next week this will all seem less confusing so I can attempt to explain what’s going on. I think I’m pretty close to figuring it out so I’m somewhat optimistic I’ll have an answer to how dot products work by next week. I have a feeling I won’t understand why they work though…

I think it’s going to take me quite a bit longer than I anticipated it would for me to get through Multivariable Calculus (100/4,800 M.P.). Considering I haven’t opened up KA in the past two weeks, plus I still have 6 more videos left to get through in the Essence of Linear Algebra playlist, PLUS even when I do get back to KA I have ~5-6 more concepts to review before I get through the first unit which is all review, I’m guessing it’ll be at least a few weeks before I get to just the second unit of MC. The silver lining is that when I finally get through MC and move onto Linear Algebra on KA, after going through this playlist I’ll hopefully (🙏🏼) be able to get through the subject relatively quickly.