I got my butt kicked this week on KA. 😔 I only made it through four videos, so I didn’t end up getting through the final section of Linear Algebra. The videos all had to do with eigenvalues, eigenvectors and eigen spaces, all of which I thought I had a decent understanding of heading into this week, however I found all the videos pretty hard to follow. I remember watching videos from other creators which made eigenvectors sound like they’re the vectors in a transformation that are scaled up or down but don’t rotate. I’m 99.99% sure that’s what they are, and I think that’s what the KA videos were saying, but the proof that Sal worked through (which was the first video I watched this week) didn’t seem to connect to that idea. The following three videos were examples of Sal working through solving eigenvalues and eigenvectors which was good practice in how to do the math, but why it worked or even what was going on didn’t really stick for me. So all in all, it was a bad week for me. BUT, I feel confident I’ll wrap my head around it all eventually. 💪🏼😤
Video 1 – Proof of Formula for Determining Eigenvalues
As you can see from my notes above, I watched this video three times but the proof didn’t sink in. The only silver lining was that it was helpful hearing Sal talk about the notation for me to make a stronger connection between the symbols and what they denote. Other than that, I didn’t get much out of this video… 😒
Video 2 – Example Solving for the Eigenvalues of a 2×2 Matrix
This video continued on from the previous one. To find the Eigenvalues of that transformation, Sal took what he was working on in the proof video and applied it to the 2×2 matrix A that you see can in the first screen shot. Even though I don’t really know what is going on or why any of this works the way it does, doing the actual math (what I think of as the how in all of this) doesn’t seem that difficult. The formula to find the Eigenvalues is det(λIn – A) = 0 which is simply multiplying λ into the identity matrix, then subtracting the column vectors of A from the lambda-identity matrix (or whatever you would call that), then finding the determinant which is just (ad – bc). To finish it off, you factor the polynomial leaving you with the solutions for the Eigenvalues. Boom. 🧨
Video 3 – Finding Eigenvectors and Eigenspaces Example
I’ll let my notes speak for themselves on this one, but as you can see at the very end of my written notes, I still don’t understand the vector column notation and how/why it works. A part form this, solving the eigenvector using the eigenvalue wasn’t too difficult.
Video 4 – Eigenvalues of a 3×3 Matrix
Again, I’ll let my notes speak for themselves here. Sal used what’s called the Rule of Sarrus method to find the determinant leaving him with what’s called the “characteristic polynomial”. As you can see in my notes, Sal then said that he assumed the solution(s) to the polynomial would be integers because he this question out of a book, and then said that assuming that was the case, the integer would be a factor of the constant, 27. (I had no idea that’s how it worked. 🤷🏻♂️) I would have had no idea how to do the polynomial long division at the end but to be fair, I also wouldn’t have known how to solve the question in the first place if I wasn’t following along with the vid.
So ya, not a great week. My plan for this coming week is to get through the final two videos of Linear Algebra and then watch at least a handful of eigenvector videos from other creators to review what’s going on. After that, I’ll FINALLY be getting started on Differential Calculus. I think I like calculus more than linear algebra, so I’m excited to get back to it. (As excited as one can get studying math anyways.) Plus, I think calculus will (hopefully) be easier now that I have a better understanding of linear algebra. I remember that linear algebra came up in Multivariable Calculus a fair amount with î, ĵ, and k̂, but I never really knew what was going on. Even though I don’t completely understand linear algebra at this point, I’m confident I’ll be able to visualize those unit vectors much better now which will hopefully make differential calculus (and multivariable calculus) easier to visualize and understand. 🤞🏼