I didn’t have a strong end to the week, but I DID manage to achieve my goal of getting through the final 12 videos in this unit, Matrix Transformations. 😮💨 It was a photo finish at the end 🏎💨🏁📸 and, unfortunately, there were a bunch of things flew over my head in the last few vids. A lot of what I found confusing had to do with notation and understanding what Sal was referring to. I explain more at the end of this post, but the confusing bits had to do with taking the transpose of a matrix (‘transpose’ means flipping a matrix so that the columns become the rows and the rows become the columns) and then finding the null space and column space of the transposed matrix and comparing it to the null and column space of the OG matrix. Because Sal was going back and forth from both matrices and they had the same elements, rows, and columns but in different places, the terminology became convoluted and difficult to follow. I think I mentioned this last week, but I feel like this is the type of math that will seem straightforward in retrospect but right now seems overwhelming and impossible to wrap my head around. Nonetheless, even though I’m not thrilled with how the week ended, I’m confident I’ll figure it all out eventually and am pumped that I made it through this unit before the New Year. 💪🏼
(In some of these videos my hand written notes do a decent job of explaining what each question was talking about so I’m not going to write an explanation of what’s going on for every video below. 🙃)
Video 1 – Simpler 4×4 Determinant
At the end of last week I tried watching and working through this video on my own and ended up getting the ERO’s wrong. This week I simply watched Sal work through it and copied the notes. As you can see in the picture of my notes, at one point Sal swapped two rows and then said you needed to flip the sign in front of the matrix from (+) to (-). I don’t remember ever learning his before so I’m still a bit confused on how or why this works the way it does… In regardless, the point of this video was that instead of finding the determinant of a 4×4 matrix by subdividing it into three 2×2 matrices and using the (ad – bc) method, you can just turn the matrix into its REF form and then find the product of the elements in the diagonal. Boom. 🧨
Video 2 – Determinant and Area of a Parallelogram
Video 3 – Determinant as Scaling Factor
I honestly don’t remember what was going on in this video. I think Sal was working through the formula for how to find the area of a parallelogram after a transformation using the given vertices. Looking at the screenshots and my notes, I don’t understand how or why the math works, but Sal’s point was that the formula to find the area of this parallelogram after the transformation was k1k2 (ad – bc) where k1 and k2 are two of the parallelogram’s vertices. 😬
Video 4 – Transpose of a Matrix
This was the first video in the final section of the unit which was titled Transpose of a Matrix. The video just went through the definition of what a transpose is.
Video 5 – Determinant of Transpose
I hope my note above makes it clear what this video was talking about and why it’s true that the determinant of a matrix is equal to the determinant of its transposed matrix. The math here was pretty hard to follow, but I at least understand what Sal was talking about even if I didn’t understand how or why the math worked. 🤷🏻♂️
Video 6 – Transpose of a Matrix Product
I’ll let my notes speak for themselves on this one, but you can see how wordy and confusing the reasoning behind the solution gets. This is what I was referring to in my intro, and this question was just the tip of the iceberg. 😵💫
Video 7 – Transpose of Sums and Inverses
Video 8 – Transpose of a Vector
AGH! I just read my note above and it’s SO confusing. I’m pretty sure it does makes sense, but I have to read it slowly and think about each part of the sentence for me to visualize what’s going on. Anyways, my note above referred to what Sal covered in the first half of this vid, but then Sal talked about something else in the second half. For that part I made a note that, “The rest of [this video] is saying if x and y are (m x 1) and (n x 1) vectors and you want to multiply them by an (m x n) matrix, A, you can take different combos of xT, yT, and AT to do so, and they will end up being equivalent.” (Whatever that means…)
Video 9 – Row-Space and Left Null-Space
From my notes:
“[This video] shows Sal finding the null space of A and noting that its column space is the first and only vector column of A. He then sets A to AT, finds rref(AT), and finds null(rref(AT)) which had one pivot entry and is the row space of A. Honestly, I don’t understand the null or column space [part of this question] and this video is confusing, but I think he’s just showing how to do it and that the Rank(A) = Rank(AT).”
(None of that makes any sense to me now, but that’s what I wrote at the time…)
Video 10 – Visualizations of Left Null-Space and Row-Space
Again, no clue what’s going on this video, but I wrote that Sal said:
“Every member of [the] column space is going to be orthogonal to every member of [the] left null space.”
And also:
“[The] column space – its orthogonal compliment is the left null space, a.k.a. the null space of the transposed matrix.”
To summarize this, I also wrote the equation ‘left null-space = xA = Null(A)’, but looking back at this, I think it may be wrong. 🤔
Video 11 – Rank(a) = Rank(transpose of a)
I thought I was going to write more notes for this video but then it got really confusing so I couldn’t. I did write that:
“I don’t understand the proof for [this video], that Rank(A) = Rank(AT). It just seems like Sal creates some hypothetical matrix, finds its RREF form, and then says, ‘since the pivot entries of the column space are a basis for A and the pivot entries are a row space for AT, they have the same dimension.’ …What?!”
Video 12 – Showing That A-Transpose x A is Invertible
This was the most confusing video of all time. I was yelling at my computer screen by the end of it as Sal way saying a bunch of gibberish that sounded something like, “since a = b then c = d, and since d = b, then a = c, and c = b so d = ” etc. 🥵 I’m sure I’m coming across as ungrateful here, but I was super frustrated and annoyed, and didn’t understand anything of what Sal was talking about. However, I did write:
“I think the point of [this video] is if you have an (m x n) matrix and the columns are linearly independent, you can construct it as a square (n x n) matrix by multiplying ATA which will end up also having linearly independent columns and will therefore be invertible.”
Again, I don’t have a strong grasp of what that means but I’m pretty sure it’s correct.
And that’s it. I made it through all the videos but clearly got smoked at the end. There’ve been times when in this scenario I’d go back and rewatch all the videos and then branch out to other YT creators to find related videos to try to better understand what’s going on, but I genuinely think that if I keep moving forward, a lot of what’s going on with transposes and how they work will sink in later on. I’m looking forward to getting started on the third and final unit this coming week, Alternate Coordinate Systems (Bases), although I have a feeling it’s going to be tough. Given that it’s the final unit of Linear Algebra, if I’m still as confused by this upcoming unit as I am right now, I’ll Google other videos on whatever I find confusing before moving on to Differential Equations. I’m starting to see how linear algebra and calculus will come together to do things like create 3D objects. I’m looking forward to FINALLY being able to bring them both together, hopefully sooner rather than later. 🤞🏼