I had another week that wasn’t super productive, and yet I still feel like I managed to learn a fair amount and get a much better grasp on linear algebra, in general. I continued this week watching the Dr. Trefor Bazett’s YouTube playlist, Linear Algebra, starting at the 29th video and finishing the week with the 36th video. In the span of six days, getting through only eight videos is pretty disappointing, but I still feel like I made some decent progress, nonetheless. I also did a good job of taking notes which I didn’t always do in KA. Taking notes has helped make a lot of the concepts sink in. Most importantly, I’m starting to really wrap my head around how matrix multiplication works which is a huge relief and makes the entire thing seem like WAY less daunting. 😮💨
Like last week, I’m just going to explain the videos one at a time in succession:
Video 1 – Finding the Matrix of a Linear Transformation
In this video, Trefor walked through each component of the rotation of two basis vectors using matrix multiplication. He showed how the trig functions with respect to theta, and the adjacent and opposite sides of the vectors relate to their new (x, y) coordinates after the transformation, and how it makes sense to place those trig values into the transformation matrix.
The second example he did showed how you can “flip” the basis vectors across the diagonal line x = y and then rotate them 90° clockwise, and how the matrices that correspond with those transformations would be written.
Video 2 – One-to-One, Onto, and the Big Theorem II
I found this video a bit confusing. I think the difference between 1-to-1 and “Onto” is that if a vector transformation leaves ALL the vectors in the same dimension (ex. R3 -> R3), then the transformation is 1-to-1. However, if some or all of the vectors go from a higher dimension to a lower dimension after the transformation (ex. if some or all the vectors go from R3 -> R2), then the transformation is “Onto”. (I could be wrong about that though.)
Video 3 – The Motivation and Definition of Matrix Multiplication
This video just went through the broad, general notation for matrix multiplication. It was helpful but because it was so general, I found it somewhat hard to follow. I kept thinking that if the matrices were written as A*B*C, then the dimension of each would be something like (3×4)(4×5)(5×1). Even though thinking through the general notation seemed vague and ambiguous, I felt like I had a solid enough grasp on how matrix multiplication works to follow along with what Trefor was talking about.
Video 4 – Computer Matrix Multiplication
Up to this video, I couldn’t understand the how multiplying two matrices by each other worked. This video helped this operation sink in for me. I understand now that you can think of the second matrix as being split up into its column vectors, multiplying each separate column vector by the transformation matrix (the matrix on the left), and then combing the resulting column vectors (the “product” vectors? 🤔) into another matrix. It takes a long time to do the math, but it seems pretty simple now.
Video 5 – Visualizing Composition of Linear Transformations **aka Matrix Multiplication**
This video helped give me a more intuitive understanding of what’s going on geometrically with matrix transformations and how they relate to the transformations of individual vectors. I didn’t bother writing any of my own notes for this video thinking it would be easier and more effective to just take screen shots and explain what Trefor was talking about in each one:
Here he begins here by showing how the green vector can be broken down into the red and yellow vectors, i.e. one step in the direction of the red vector and two steps in the direction of the yellow vector.
This shows the first transformation the video will look at.
This shows the second transformation the video will look at. Notice that even after both transformations have been done, you can still get to the end of the green vector by going across one red vector and two yellow vectors. This means that all the vectors have 1) stayed proportionate to each other and 2) haven’t changed positions relative to each other. (As far as I understand, those two points are what linear algebra is all about.)
This shows that the first transformation will be denoted with the capital letter “T”.
This shows that the second transformation will be denoted with the capital letter “S”.
This shows that the linear combination of the transformation T (in the center of the equation) is just another way of writing out the matrix multiplication of the transformation (on the right side of the equation). I now understand this as being basically the same thing as factoring and expanding polynomials in basic algebra, except it’s the linear algebra version of it.
Shows the same thing for the transformation S.
This final screen shot shows the algebra for the matrix multiplication ST (in that order!). Note that T has already been broken apart into its column vectors in the top equation. (Not sure if I’m saying that correctly or if it’s even true.)
Video 6 – Elementary Matrices
This was a very interesting video. It showed how instead of writing out things like R2 <-> R1 to denote switching Rows One and Two in a matrix, you can multiply a matrix by an altered identity matrix to achieve the same thing. I definitely still need practice with this to quickly be able to interpret what transformations are taking place, but I was happy that I somewhat intuitively understand what Trefor was talking about here. (Before he gave the solution to the final transformation, adding K times R1 to R2, I paused the video and figured out that you would place a k in the first column of the second row. 💪🏼)
Video 7 – You Can “Invert” Matrices to Solve Equations…Sometimes!
In this video, Trefor was emphasizing that 1) if a matrix has an inverse, the matrix multiplied by its inverse equals the identity matrix, and 2) linear algebra is WAY easier (apparently) if a matrix in question has an identity matrix. (I’m also pretty sure there’s a typo in this video in the first box where it says “AA-1 = A-1 = I” which should actually say “AA-1 = A-1A = I”.)
Video 8 – Finding Inverses to a 2×2 Matrices is Easy!
Here Trefor was just going through the formula to find the inverse matrix of a 2×2 matrix. I’ve done this many times before on KA, but it was still good review. It was also interesting hearing him say that the denominator of 1/(ad-bc) is known as the “determinant” which I definitely remember from calculus but I don’t understand how it relates to the inverse matrix yet. (Partly because I don’t understand why the formula for the inverse matrix works in the first place.)
And that was it for this week. Like I said, not my best week, but it also could have been worse. I’m sad that I’m not making quicker progress through this playlist, but I feel like I’m learning a lot and in the big picture, I’m sure these videos will quicken my progress in the long run. At this point, it’s hard for me to imagine ever not studying math or reaching my goal of getting through the MATH section on KA. I know I’ll get there, but it might take another year. It’s not the end of the world, but it feels like I’m just treading water, and it’s hard to stay motivated. I didn’t come this far just to give up though! On to Week 265… 🙃