I essentially made zero progress on Multivariable Calculus this week, but I still had a fairly productive week and am pretty happy with what I was able to learn. My goal was to get through the next few sections of the first unit in MC and I didn’t even make it through the first one, Dot Products. I opened up the article for that section and was completely lost so I Googled dot products and found a video from 3Brown1Blue. That video was the nineth video of Grant’s Essence of Linear Algebra playlist. I couldn’t understand the video on dot products so I decided to start from the beginning of the playlist and watch all the videos leading up to it hoping that I’d be able to understand after getting through the first eight videos. My entire week was spent going through those videos which gave me a MUCH better understanding of vectors. I’m bummed that I didn’t make any real progress through MC but am definitely happy that I have a better intuitive grasp on vectors. 👍🏻

Before I got into the 3B1B linear algebra playlist, on Monday I made a quick note on how Pythag’s Theorem works with 3D triangles:

As I mentioned at the end of my last post, this still blows my mind that a² + b² + c² = d² mainly because for years (decades if I include the first time I learned it in middle school) I’ve been saying a² + b² = c². I ran out of space on the page from my note above and didn’t want to start a new page, so it maybe isn’t very clear to understand. To sum it up, line C is the hypotenuse of the triangle on the base of the pyramid. It’s length is 2² + 5². Line C is also the opposite or adjacent side of the vertical triangle depending on how you look at it, meaning that its value plus 8² equals d² and therefore, 2² + 5² + 8² = d².

I started the Essence of Linear Algebra playlist on Wednesday morning. This was the first note I took down from the first video in the playlist:

I thought it was interesting when Grant explained that there are different ways to denote and think about vectors but they all mean the same thing. It turns out that being able to interchangeably think of vectors as arrows, and as rows and columns of numbers makes them a LOT easier to understand. The third type of notation, where there’s a letter with a small arrow over top of it, didn’t seem as useful as the first two types of notation. The ladder type of ‘math’ notation was used in the videos as more of a placeholder or variable to denote the vector but doesn’t give you any explicit information about the vector itself.

In the second video Grant explained how to think of vectors from the physics perspective using arrows. He explained that if you’re thinking about one vector, you should visualize a single arrow but it you’re thinking about multiple vectors it’s often best to think about each vector as a dot on a cartesian plane:

In the note above I said that multiple vectors should be thought of as points on a 2D plane but I’m sure hat’s also true for vectors in a 3D space, as well.

On Thursday I wrote down the definitions of two terms the come up with matrix transformations which were Linear Transformation and Linearly Dependent:

In my opinion, the more interesting of the two definitions was linear transformation. This is when a certain vector is transformed by some sort of matrix but in a way where all the points/grid lines of the space that the vector is in get adjusted to the same scale as the vector itself. The simple way to think about this is that if a vector gets transformed and none of the space around the vector gets curved, it’s a linear transformation.

On Friday I watched and made notes on the 6^th video in the playlist which made everything WAY easier to understand. It explained what are known as the basis vectors (which I think of as ‘unit‘ vectors) which are three individual vectors that are exactly one unit in length in either the x, y or z direction. The notation for each of them respectively are pronounced and denoted as; “i-hat”, î = (1, 0, 0), “j-hat”, ĵ = (0, 1, 0), and “k-hat”, k̂ = (0, 0, 1). I now understand that the columns in a 3×3 matrix correspond to the coordinates of î, ĵ and k̂ and the rows denote where the x-, y- and z-coordinates of î, ĵ and k̂ would be mapped to. I’m not sure if that makes sense, but here’s a few pages from my notes that hopefully make it more clear:

It turns out that with this type of matrix transformation, because it’s a linear transformation all other points within the 2D or 3D space will be transformed to the same ratio. This means that you ONLY have to think about how î, ĵ and k̂ get transformed to understand how the entire space gets transformed. This makes it all seem much simpler to me.

Here’s a page from my notes that explains how in a 2×2 matrix the top left and bottom right numbers stretch the vectors and the bottom left and top right numbers rotate the vectors:

Finally, here’s one more page from my notes that more-or-less goes talks about the same concept but adds the third vector, k̂:

By the end of the week I’d made it back to the video on dot products which got me started with this playlist in the first place. I watched half of it and was still a bit confused, although it did make more sense watching it the second time than it did the first time. Like I said at the beginning, I’m sad I didn’t actually make it through anything on Multivariable Calculus (100/4,800 M.P.) but I’m sure that what I learned about vectors, especially better understanding the basis vectors, is going to make things a lot easier for me when I get back to MC. There are 16 videos in the Essence of Linear Algebra playlist and I’m planning to go through all of them this coming week. My goal is to get through all 16 videos, then rewatch the video on dot products and hopefully actually understand what they’re talking about, and then get back to KA and work through the dot product article. This once again feels like it’s taking me a really long time to get anywhere, but I can tell I’m making progress which I’m happy about.