All in all, it was another “meh” week for me. I only made it through six videos on KA but they were all around 20 minutes long. Even though I didn’t watch many videos, I’m sure I made progress with linear algebra, in general. I’m getting better at being able to visualize what the math represents. I can see through the notation to the geometry of what’s going on and am much quicker at interpreting what’s happening. A metaphor would that it’s as if I can read entire sentences at a glance now as opposed to having to read each word individually and stitch together the meaning of each word to understand the sentence. I know that I’m still miles away from really having linear algebra figured out, but the fact that I can understand what the notation represents – from matrices to vectors to matrix addition and multiplication – there’s much less friction for me watching and working through the videos. So, ya, not a great week, but still making progress nonetheless! 😬
Video 1 – Introduction to the Inverse of a Function
This video was just under 20 minutes long and seemed like it kept going around in circles. As always, I could have missed something entirely, but it seemed like Sal kept reiterating the three points in the picture of my notes above over and over. I’m sure he was just doing this to make each point as clear as possible, but it was the type of video that (in my opinion) could have been about half the length. 🤷🏻♂️
(That sounds super ungrateful. I did find it useful going through the notation for inverse functions and watching him draw diagrams of functions mapping vectors from a domain to a codomain. This is the type of thing really helped me be able to visualize what’s going on with matrix multiplication.)
Video 2 – Proof: Invertibility Implies a Unique Solution to f(x) = y
In this video, Sal spent 23 minutes proving that if a transformation takes ALL the values in a set, ex. X, to unique/individual value in another set, Y, then the transformation has an inverse. Also, the composition of the function with its inverse equals the Identity Matrix:
- f(f-1(x)) = I(x)
Takeaway: An invertible function implies a unique solution.
Video 3 – Surjective (“onto”) and Injective (“one-to-one”) Functions
This was one of the shorter videos of the week but was very helpful. In it, Sal talked about a two terms and some notation used with transformations. I’ll let my note above speak for itself, but I’m pretty sure this is the first time in my 34 years of being alive that I’ve ever heard the word “surjective”. Apparently its etymology is from a Latin word (big surprise) that means “throw in”, as in all the values in a set X get ‘thrown in’ to the another set, Y.
Video 4 – Relating Invertibility to Being Onto and One-to-one
In this video Sal did a quick diagram of what surjective and injective transformations do and noted that for a transformation to have an inverse, it must be both surjective and injective.
Video 5 – Determining Whether a Transformation is Onto
Again, this was another video where Sal worked through the algebra and made a diagram of what it means for a transformation to be surjective. I know there was at least one other important point to this video, but I can’t remember what it is now. 😔 I think it had something to do with rows of 0’s meaning the transformation had parts of the domain that ‘collapsed’ in the codomain, i.e. the number of dimensions (a.k.a. the “rank”) went down. I’m not sure about that though…
Video 6 – Exploring the Solution Set of Ax = b
This was the final video I watched this week and it worked through an example of a matrix transformation that wasn’t surjective. (I think. 🤔) In this example, the transformation sends a bunch of the vectors in the domain, in R2, onto a single line in the codomain, R1. Because of this, I’m pretty sure this particular transformation is NOT surjective. Once again, this video was really helpful for me to visualize what’s happening with matrix transformations.
And that was it for this week. I didn’t get too much done, but I’m glad I have a better grasp on the notation and terminology used. Plus I’m happy with the notes I took, so I’ve at least got that going for me which is nice. I have a little less on my plate this coming week work-wise, so I’m hoping to make it through ~10 videos. I have 28 videos left in this unit so if I can manage to get through 10, that would set me up well to finish off this unit before the start of the New Year. Even if I do, I’ll still be well behind where I thought/wanted to be at this point, but I’d still be happy regardless. I’ll get to the end of this Math journey eventually, but I may be an old, decrepit man when I do. 🧓🏼