It wasn’t a great week for me on KA. 😔 I only made it through five videos and most likely spent less than five hours this week working through them. They weren’t the easiest videos to understand, and I watched a few of them more than once, so it’s not like I didn’t put any effort in. But, nonetheless, considering how badly I want to get through Linear Algebra, I could and should have done more. Part of the reason I didn’t get more work done was because I’ve been spending a lot of time working on this side project that I’ve mentioned in a few posts, and I’m still also having health issues which has been slowing me down. I hate making excuses, but between studying KA, this side project, and my health, (not to mention my actual job) I’ve been stretched pretty thin lately. BUT, I’m making progress and know that I’m going to get through LA soon. 😤
Video 1 – Changing Coordinate Systems to Help Find a Transformation Matrix
I will let my written notes speak for themselves on this one.
Video 2 – Introduction to Orthonormal Bases
This was the first video from the second last section of Alternate Coordinate Systems. As the title of the video, Introduction to Orthonormal Bases, would suggest, it went through the definition of a what an orthonormal basis is. What I wrote in my notes was:
- Definition of Orthonormal:
- “Ortho” = orthogonal
- “Normal” = Length of 1
- Think unit vectors (î ĵ k̂) but they don’t have to ON the axes
- All vectors in orthonormal bases are linearly independent
- All basis vectors dotted with themselves equal 1, and all basis vectors dotted with another basis vector equal 0. (Because they’re all orthogonal to each other.)
Video 3 – Coordinates with Respect to Orthonormal Bases
Again, I’ll let me notes speak for themselves here.
Video 4 – Projections Onto Subspaces with Orthonormal Bases
I watched this video twice and found the math hard to follow. I believe the point was to say if you have an orthonormal basis, it’s much easier to find a projection of a vector onto it than of the basis ISN’T orthonormal. If the basis ISN’T orthonormal, you have to use the formula projV(x) = A(ATA)ATx, however if the basis IS orthonormal, the (ATA) turns into the identity matrix leaving you with projV(x) = A(I)ATx which is just projV(x) = AATx.
Video 5 – Finding Projection Onto Subspace with Orthonormal Bases Example
As you can see, this video worked through an example of what Sal was explaining in the previous video. I definitely don’t completely understand everything that’s going on, but the math in this example doesn’t seem too difficult and I think I understand the gist of what’s going. At the very least, I feel much more comfortable with the notation and the operations used, plus I can 100% visualize what’s going on with the plane create by the basis vectors and how the projection of a vector onto that plane would look and relate to it.
And that was it for this past week. Like I said, it was disappointing that I didn’t get more work done, but clearly I did get some work done and made some headway understanding Linear Algebra. I don’t feel nearly as overwhelmed and out of my depth now with LA as I did a few weeks ago. I feel WAY more comfortable with conceptualizing matrix transformations on vectors and using matrix multiplications, dot products, RREF, etc. I have 12 videos left to get through in LA, so if I actually make a decent effort this coming week and get through six or more vids, there’s a good chance I can get through LA by the end of Week 285. Then it’ll be onto Differential Equations which, a part from redoing the Multivariable Calculus Course Challenge, will be the FINAL THING left for me to do in the KA Math section…… 😳 And it will only have taken me six years.