FINALLY! After what feels like forever, I finally had a week of studying where I feel like I got a lot done and actually made some decent progress through KA. 😮‍💨 It took me until Thursday to wrap up the 3B1B Essence of Linear Algebra playlist and by the end of it I felt like I had a WAY better grasp on vectors than I did going into the playlist. Nonetheless, I still feel like I’ve barely scratched the surface when it comes to linear algebra and vectors. When I got back to Multivariable Calculus on KA, I breezed through the five articles in the second section of the first unit, Thinking About Multivariable Functions. The articles were pretty short, but the main reason why I got through them so quickly was because the Essence of Linear Algebra playlist went into much greater detail than what was covered in the articles. Having spent the last four weeks watching the vids from that playlist, getting through the articles so quickly definitively felt vindicating having spent so much time on watching them. I’m sure that having a better intuitive grasp on what was covered in those vids will make the next few sections/units in MC much easier to understand. 

Before I started back on KA, I rewatched the dot product video one last time. I made a note about a concept known as “duality” which Grant emphasized the importance of. Here’s a screen shot from the vid when Grant was talking about duality:

I’m definitely not going to attempt to explain why it works (because I don’t completely understand it), but the concept of duality is that a vector written on an (x, y) grid can be broken down into its x- and y-components and rewritten on a number line and it’s the same thing. I think I made a note about why this is true a few weeks ago where Grant draws a diagonal number line through the origin of an (x, y) grid and uses symmetry to show how different points on the grid can be brought onto the number line and they’re somehow equal in value. I can’t explain it but can somewhat visualize it, so I have that going for me which is nice. Grant says that the important thing to know is that points on an (x, y) grid can be rewritten as two vectors (I think is the right word?) on a number line, which is what duality means, and that this is very useful to understand going into linear algebra and I’m guessing into MC.

The five articles I got through on KA were titled Dot Products, Cross Products, Matrices-Intro, Visualizing Matrices and Determinants. The only note I made on dot products this week was that Grant said, “the dot product is a very useful geometric tool for understanding projections and for testing whether or not vectors tend to point in the same direction.”

The next article, Cross Products, took me a day to get through. I believe a cross product of two vectors is when you slide each one along the other to create a parallelogram. Here’s an example:

I don’t really remember how it works, but here’s an example of how to find the cross product for a 3×3 matrix:

Question 1

I don’t really understand what a 3×3 cross product does. I think it takes two (x, y, z) vectors and gives you a third (x, y, z) vector which is perpendicular to the OG vectors. I just googled it and found that the magnitude of that new, third vector equals the product of the magnitudes of the two OG vectors multiplied by the sine of the angle between them. I don’t know why this is useful to understand.

In the next article, Matrices-Intro, I wrote down a bunch of important terms/definitions:

After I got through those terms, I worked on a few questions that dealt with matrix multiplication. The key insight I had here was that when multiplying matrix A by matrix B (written as AB) to get matrix C as the product, the location of each element in C corresponds with the same element in the rows of matrix A and the same element in the columns of matrix B. (I’m sure there’s a better way to say that but it’s hard to put into words.) There’s more to it than that, but I’m not going to explain the technique here. BUT, here are two example questions:

Question 2

Question 3

The last thing I worked on in this section was how matrices move vectors. I found the way that KA described it in the article with their diagrams was a bit more enlightening and helped me grasp the intuition of how/why it works. Here are a couple of screens shots from that article and a note I took at the very end:

I still don’t fully understand how the operation (not sure if that’s the right way to use that word) of matrix/vector multiplication works. The how of it, so to speak, (i.e. the method) doesn’t seem too difficult to follow but the why of it, as in why it actually works, doesn’t make sense to me. 🤔

I finally moved on to the next section, Visualizing Scaler Valued Functions, on Saturday and made it through the first of four videos. I felt like I had already worked on/learned so much this week that I didn’t want to try to learn anything more and overwhelm myself so I didn’t make too many notes on this video. The one takeaway I had from that video was simply how to orient the (x, y, z) Cartesian coordinate system. I always assumed that the x- and y-dimensions would stay the same and that the positive z-dimension would come towards me and away from me for negative z-values. I looked this up last week and found conflicting answers, but the way it’s described on KA is that you tip over the y-dimension so that the x- and y-axes are “laying flat” and the z-axis is vertical with positive z-values going up and negative z-values going down:

Having gone through all of that this week, I still didn’t earn any mastery points in Multivariable Calculus (100/4,800 M.P.) so it says I’m still only 2% of the way through. I have 12 videos and eight articles left to get through in this unit (none of which will earn M.P. 😒) so I’m guessing it’ll be two-ish weeks before I move onto the next unit, Derivatives and Multivariable Functions. I’m looking forward to getting to it but I’m sure the review I’m doing now will make things a lot easier to understand going further into M.C. Even still, I’m getting a bit impatient and just want to get through it. 😤