I made it through seven videos this week from Dr. Trefor’s Linear Algebra playlist. I wish I would have made it through more, but the notes I took were all pretty legit, plus I feel like I actually made a ton of progress understanding a bunch of the notation used in LA and had a better general idea this week of what Trefor was talking about in the videos. I’m pretty far from having it all 100% figured out, but I now feel like I have a decent grasp on what’s going on with some of these concepts. This makes me think that understanding everything else will (hopefully) be more intuitive for me, going forward. I also got a bit of practice this week with actually DOING matrix multiplication, which is always helpful to make the concepts sink in. So overall, I’m pretty happy with how this week went, although I’m disappointed that it’s taking me so long to get through this playlist.
Once again, here are screen shots from all the videos I watched this week and notes I wrote out for a few of them:
Video 1 – What Exactly are Gird Lines in Coordinate Systems
In this video, Trefor talks about grid lines saying that…
The equation at the top of this screen shot is the equation for a line running parallel to the y-axis @ x = 2. The fact that it’s @ x = 2 is denoted in the equation as 2e1. Where it then says te2 | ∈ R, that means that the grid line runs parallel to the basis vector e2 (i.e. it runs parallel to the y-axis) and can go infinitely ‘up’ or ‘down’ given that ‘t’ is a member of the real numbers, R. (That wasn’t a great way of phrasing it, but I think the gist of what I just wrote is correct.) At this point in the video, Trefor also noted that all grid lines fall on integers.
“Grid systems are all the integer linear combinations of the standard basis vectors.”
If you take a non-SBV…
You can still make a grid system, and…
Get to a spot with the same “rules” as when doing it with SBV.
Knowing this is (apparently) useful for when, for example, you’re finding a plan in R3. (Or something like that…)
Video 2 – The Dimensions of a Subspace | Definition + First Examples
Video 3 – Computing Dimension of Null Space & Column Space
I found the how of this video (as in how to DO the math) pretty difficult. I’m glad to say that I understand the matrix and the two equations at the top of this screen shot, but I’m still struggling to understand the vector notation at the bottom of the screen shot. (I don’t even know if it’s called “vector notation”… 😒)
HOWEVER, since there are two free columns, x2 and x4, that means the dimension of Null(A) is 2. (🤔 … 🤷🏻♂️)
This Screen shot restates what I just wrote above.
Trefor wraps up this vid by saying that the number of columns with “leading-1’s” determines the dimension of Col(A) (which I think means it determines the dimension of the matrix, i.e. whether the matrix is in R2, R3, etc.).
Video 4 – The Dimension Theorem | Dim(Null(A)) + Dim(Col(A)) = n | Also, Rank!
(See hand written notes below for explanation of this screen shot.)
The green line equals the Dim(Null(A)) before the transformation.
This shows the Dim(Null(A)) DURING the transformation.
This shows the Dim(Null(A)) after the transformation. (As in, all the vectors along the green line before the transformation go to the dot at the origin. Well, actually, they all collapse to 0 after the transformation, but they do it on the origin, or something like that.)
This screen shot doesn’t show it, but here Trefor was talking about how the entire grid collapses down to the x-axis during the transformation which (I guess?) is the Col(A).
(Basically, all the vectors BEFORE the transformation that will END UP going to 0 are a part of the “Null Space” and all the vectors that collapse from a higher dimension, R2, into a lower dimension, R1 on the x-axis, AFTER the transformation are the “Column Space”. I think…)
Video 5 – Changing Between Two Bases | Derivation + Example
Q. If you have two different looking grids (again, a.k.a. basis), how can you find that same vector relative to the standard basis in another grid?
(Looking back at the screen shot and at my notes, I don’t really know what’s going on here… But the following is exactly what I wrote in my notes.)
Note: b1, b2, … bn are BASIS column vectors (I think), like [1, 0, 0], [0, 1, 0] and [0, 0, 1], so s1, s2, … sn are the weights/coefficients for x.
(… whatever that means…)
I don’t understand what “P” denotes in PB(x)B. I think it means something like “x inside of B is going to equal some set of weights (coefficients), [s1, s2, … sn], multiplied by the basis vectors in B, [b1, b2, … bn]. THAT is what PB(x)B means.”
(Uhh, what?)
This is the notation to switch from one set of bases (think “grid system”) to another set.
This is an example of how to do the math to find the transformation matrix.
This shows how the transformation would look visually on a grid.
Video 6 – Visualizing Change of Basis Dynamically
This was the most useful video for me this week. Again, I’m just going to rewrite exactly what I wrote in my notebook underneath each screen shot.
The vector x inside the B grid system (a.k.a. the B basis) looks like this.
The B basis from the perspective of the Standard Basis (SB) system looks like this, therefore x in B in SB looks like this.
(Hypothesis: It may be that PB(x)B is the linear equivalent of saying f(g(x)).)
If you keep x in the same spot but change the bases from B from perspective of SB to C from perspective of SB, the C basis overlaying the SB basis looks like this.
If you go from where x was from SB perspective to where x is from C’s perspective, x looks like this.
(Rereading that, that’s very confusing but I think it’s actually correct.)
Nested notation: x inside of B; B from perspective of SB; SB from perspective of C.
(The way this is written in the screen shot definitely reminds me quite a bit of compound functions, like it’s the equivalent of saying f(g(h(x))).)
I was trying to think through what was going on in this video and came up with a currency metaphor to help me wrap my head around it. I asked CGPT if I was on the right track and this is what it said:
Video 7 – Example: Writing a Vector in a New Basis
I think my notes do a decent job at explaining this one so I’m not going to elaborate.
I’m now finished 60 of the 82 videos in this Linear Algebra playlist. I would be SO happy if I could get through the rest of them in the next two week, but I have a feeling that won’t happen. I’ll have to get through the videos ~30% faster than I have been for the last few weeks if I’m going to do it. So, it’s not impossible but it’ll be difficult. One thing I’ve got going in my favour is that I’m pretty enthusiastic right now and motivated. I’m SO close to getting through this playlist, and close to finishing off the MATH section in KA (relatively speaking, anyways). I’ve said this before but getting through the MATH section of KA will be a big achievement, in general, for me but maybe more importantly, it’ll be a huge win for me symbolically. Five years ago I told myself that if I could learn calculus, I could do anything. There are a confluence of challenges happening in my life right now, good and bad, and they all have a lot of potential to make my life better, but they’ll all be difficult to manage and overcome. When I finish the MATH section of KA, I feel like it will represent the end of one chapter and be the beginning of a new and exciting chapter of my life. I’m SOOO close the finish line. Time to get through this LA playlist! 😤 💪🏼