This was one of the best weeks I’ve had on KA in a long time. 💪🏼😤 I only made it through six videos but a lot of key concepts clicked into place for me this week. It started on the first video I worked through which gave a pretty straightforward example of changing a vector from the standard basis (where the basis vectors are unit vectors) to a non-standard basis. Sal then explained that the change of basis matrix used to go from the standard basis to the non-standard basis – also, I’m pretty sure “non-standard” isn’t the correct way to phrase it – is typically denoted with C and the inverse with C-1. These matrices are not the same as transformation matrices. I’ll explain below, but the way that I understand it is a transformation matrix will move a vector from one set of coordinates to another, whereas a change of basis matrix moves the grid behind the vectors (which, as a result, does ‘change’ the vectors coordinates, but it’s not exactly the same thing as a transformation matrix). Anyways, the point I’m trying to get at is I think I FINALLY got over the hump of understanding what’s going on with 1) the notation, and 2) visualizing/conceptually understanding what is happening with these transformations. 😮💨
Video 1 – Coordinates With Respect to a Basis
I’ll let my notes speak for myself on this one, but I think I screwed up what I was saying at the very end. I initially wrote it as a’s coordinates inside of B were [8, 7] but I’m pretty sure the right way to say it is a’s coordinates inside of the standard basis are [8, 7].
Video 2 – Change of Basis Matrix
This video helped clarify a lot of things for me. It confirmed that there are subspaces that, for example, are made up of 3D vectors but, in a sense, only span 2D if they create a plane in 3D. (I think of an infinitely large piece of paper in 3D space that is slanted so it’s not laying on the xy-plane but it still has a z-dimension.) The point of this video is to say that in such a scenario, if you have a vector in/on the subspace (i.e. the “piece of paper”) written in the subspaces 2D coordinates, [x]B, you can find its 3D coordinates (a.k.a. the standard basis coordinates) by multiplying that vector written in [x]B by the change of basis matrix. In this same type of example, if you have a vectors standard basis coordinates and you KNOW that it is in/on the subspace, you can use rref on the change of basis matrix augmented with the vectors S.B. coordinates to find its coordinates with respect to the subspace.
(I don’t know if I’m explaining this properly, but I think I understand what’s happening. Praise the lord. 😭)
Video 3 – Invertible Change of Basis Matrix
I think the example in my notes from this video may be a bit confusing. What I’m showing is that if there’s some square change of basis matrix, C, then there will be an inverse change of basis matrix, C-1, that will take a vector written in S.B. back to the non-standard basis.
Video 4 – Transformation Matrix With Respect to a Basis
I’m not 100% confident that what I wrote in my notes above is precisely correct, but I think it’s pretty close, if not. Also, if the diagram at the bottom of my notes is confusing, there are three more (hopefully less confusing) diagrams detailing the same thing in my notes in the next video.
Video 5 – Alternate Basis Transformation Matrix Example
This is what I was talking about in my intro of this post. I remember working through this type of thing when I was going through Dr. Trefor Bazett’s series on linear algebra and somewhat understanding what he was talking about but not nearly as well as I do now. I think part of the reason why it’s more clear to me now is because of the very first video that I worked through this week. The example in that video showed a simple example of how x written in S.B. was [8, 7] and x written in B, [x]B, was [3, 2]. There’s a difference (I guess?) between a change of basis matrix, C/C-1, and a transformation matrix, A and D, in the example above. To be fair, I haven’t completely wrapped my head around the nuance between the two types of matrices as they seem somewhat similar to me, but like I said at the start, I think I’m over the hump of understanding it all which is a big relief.
Video 6 – Alternate Basis Transformation Matrix Example Part 2
What I wrote in my notes for this video was:
“Sal works through an example based on [previous video] using x = [1, -1] and the values of C, A, D, and C-1, and shows that x = [T(x)]B going in both directions. He finishes the video by saying the reason why you want to change the basis of x to [x]B (using C-1) is because D’s transformation matrix contains 0’s – it’s a Diagonal Matrix – which makes it was easier to compute that A’s matrix which has no 0’s.”
And that was all the videos on KA that I made it through this week. I did however watch two videos from a new channel, Visual Kernal, which were SUPER enlightening. The channel has a grand total of four videos and I watched the first two last week and then the final two this week. The videos do an incredible job of showing examples of matrix transformations, explaining the three types of transformations (rotations, reflections, and sheers), and explains key concepts, orthogonal vectors, eigen vectors etc., in a very simple way. Here are the four videos:
And that’s where I’m going to leave it for this week. I only have 17 videos left to get through in Linear Algebra which I’m pumped about. I should be able to get through them in at least three weeks but hopefully only two. Unfortunately the next video is 30 minutes long (😑) so two weeks might not happen if all the videos are similarly long. In any case, unlike last week I now feel WAY better now about linear algebra, in general, and think I’m close to understanding how and maybe even why it all works. It feels very similar to how I felt when I finally understood (you know what’s coming…) the unit circle in Trig. Given how frustrated I was trying to figure that out and how satisfying it was when I finally did, I’m really hoping that I’ll have that same eureka type of moment happen here with linear algebra. As always, fingers crossed! 🤞🏼