It was another mediocre week for me working through the Matrix Transformations unit in Linear Algebra. My goal was to get through ten videos but I only made it through eight. If my math checks out, that means I made it through 80% of the videos I wanted to, which isn’t too bad, but I didn’t understand a lot of what was being talked about in the first few videos. 😒 I somewhat redeemed myself at the end of the week and learned the proof of the determinant of a 2×2 matrix and actually understood it. 😮‍💨 The determinant is 1/(ad – bc) which I’ve seen a million times before but never knew where it came from or its derivation, so working through that proof was pretty satisfying. All in all it was one of those, “could have been better but also could have been worse” kind of weeks.

Video 1 – Matrix Condition for One-to-One Transformation

This is an embarrassing way to start this post, but I watched this video twice and didn’t understand it at all. I wrote in my notes, “I literally have no idea what’s happening in the 7^th vid. 🙁 ”. I’m certain it has to do with determining whether or not a transformation has an inverse, but that’s about all I was able to take away from this one.

Video 2 – Simplifying Conditions for Invertibility

The main points of this video were that 1) for a transformation to have an inverse it must be a square (n x n) matrix, and 2) using ERO’s on the transformation, you must be able to turn it into the Identity Matrix.

Video 3 – Showing That Inverses are Linear

This video was 21 minutes long and the entire thing was about proving that an inverse transformation from a given linear transformation must also be a linear transformation. To prove this you have to show that 1) a transformation applied to the sum of two vectors equals the transformation applied to each vector individually and the sum of their outputs, and 2) scaling a vector before and after a transformation will result in the same output vector. (Also, I’m not sure if the phrase “output vector” is a thing, but hopefully that makes sense.) The notation you’d use to express these requirement is the same as usual apart from there being a “-1”  in superscript on the T:

• T^-1 (a + b) = T^-1 (a) + T^-1 (b), and
• T^-1 (ca) = c T^-1 (a)

Video 4 – Deriving a Method for Determining Inverses

This was the first video from a new lesson section titled Finding Inverses and Determinants. Sal’s point in this vid was that if turn a given matrix into its rref form and applied the same EROs to the identity matrix, the Identity Matrix would end up as the OG matrix’s inverse. (This concept is what’s boxed off at the end of my notes above.) Sal applied each step to the two matrices individually in this vid, but it turns out the the more succinct way of doing it is by augmenting the OG matrix with the Identity Matrix and trying to use ERO’s to switch the OG matrix into it’s rref form, a.k.a. the Identity Matrix.

Video 5 – Example for Finding Matrix Inverses

This is an example of what I was talking about in the fourth video. This type of math isn’t too difficult which makes it as enjoyable as math can be, but there are a lot of steps to it which makes it easy to make “careless mistakes”, as Sal often says.

Video 6 – Formula for 2×2 Inverse

This video showed the proof for the determinant of a 2×2 matrix. I find this algebra pretty hard because you’re multiplying and cancelling out so many letters and so it gets confusing. Plus the operations denote rows with R­­­_n which just adds more letters and another layer of complexity. I also find the first ERO, aR₂ – cR₁ = R₂, used to zero-out the bottom left element confusing, although I do understand how it works. (But just barely. 😵‍💫)

Video 7 – 3×3 Determinant

In this vid, Sal didn’t show the algebraic proof for the determinant of a 3×3 matrix, but just showed how to use it. I’ve used this formula a lot in the past so it wasn’t a difficult video for me to follow along. What stuck with me the most (which you can see at the boom of my note above) is that using two parallel lines on either side of the 2×2 elements denotes the determinant of that 2×2 matrix.

Video 8 – n x n Determinant

This video takes what Sal talked about in the previous two videos, finding the determinant for a 2×2 matrix and a 3×3 matrix, and extends it to an (n x n) matrix. In the video, Sal worked through an example of a 4×4 matrix, but apparently you can use the same strategy on any size of (n x n) matrix. I’m trying to think of how to put the technique into words but I literally can’t think of how to phrase it, so hopefully my notes above make it somewhat clear. 🫤

(I’m pretty sure the fact that I can’t put it into words means I don’t really understand it, which is true, but I do think I have the gist of what’s going on figured out. So I’ve at least got that going for me which is nice.)

Aaand that was it for this past week. Like I’ve said the last few weeks, not terrible but also could have been better. I have 20 videos left to get through in this unit and 15 days left in December. I definitely think it’s possible that I can get through the remaining videos before the end of the month, especially since I’ll be off work beginning on the 24^th, but I’ll have to get through them at a quicker pace than I’ve been getting through them recently. There’s also no unit test which will speed things along, although I do miss doing exercises, in general, since they make a big difference helping the concepts sink in. But in any case, I’m happy to still be making progress and feel like I’m actually understanding and retaining a bunch of what I’ve been going through lately, so even though it’s taking me WAY longer than I thought it would, I’ll get there eventually.