I had another very successful week working through KA! By Friday I made it through the 7 videos, 2 exercises, and unit test in Matrices and then got through all 13 videos and 8 exercises in Probability and Combinatorics by the end of Saturday. It took me awhile to remember a lot of what came up in Matrices but, as is often the case, once I reviewed the concepts that came up they actually seemed clearer and more straightforward than they did when I first learned about them. Everything that came up in Probability and Combinatorics was a breeze which was a relief. The videos were all ~2-4 minutes long and the exercises were rudimentary. Considering I initially thought it’d take me until September to get through Precalc, I’m really happy that I only have the unit test in Probability and Combinatorics left to get through before I can get started on Calculus 2. WOO!! 🥳

The majority of my week was spent working through matrix multiplication questions. Here’s a page from my notes that shows a few definitions from matrix multiplication and a basic example below them:

As far as I understand it, a vector is just an (x, y) coordinate on a Cartesian plane and a matrix transformation will scale/move the vector to another set of (x, y) coordinates. In the example above, the vector’s coordinates were (3, –1) and got moved to (–10, 3).

In one of the first videos I watched from Matrices this week, Sal said something that helped me understand how matrix multiplication works. Here are two pages from my notes that explain:

On the second page in the second line down from the top, you can see that it says a[0, 5] + b[2, –1]. What Sal said that helped me think through matrix multiplication was that if you’re multiplying a vector by a matrix transformation such as [0, 5 | 2, –1] * [a, b], you can think of the product as being “a groups of [0, 5]” and “b groups of [2, –1]”. There’s nothing mind blowing about thinking it through this way, but it somehow seems to simplify the process in my mind. The initial question int he pages above, however, was actually asking me to multiply two 2×2 matrices together, matrix A and matrix B, which Sal described as the product of “B of A”. To get to the solution, you use the same technique as I just described above but you use it twice. I’m finding it hard to put into words how the method works but if you look at the bottom half of the second page above you can see which A value gets multiplied by which B column and how to add them together.

The following are 3 example questions I worked through with the first two being from videos and the third from an exercise:

Question 1

Question 2

Question 3

On Thursday I watched a few videos about matrix determinants. The determinant of a matrix tells you how much a vector (or any matrix that’s 2×2, 3×3, etc.) will be scaled by from the determinant’s corresponding matrix transformation. For example, if you’re given a 2×2 matrix transformation and you determine that the determinant of that matrix is 5, that means if you apply that transformation to a vector, the vector’s length will increase by 5x. However if you apply that same matrix transformation to a 2×2 matrix, the matrix’s AREA will increase by 5x and the volume from a 3×3 matrix would increase by 5x.

The formula to calculate a determinant is as follows:

• Matrix A:
  • [a, c | b, d]
    • (Note: Matrix A is a 2×2 matrix where a and c make up the first column and b and d make up the second column.)
• Determinant of matrix A:
  • Det (A) = |(a*d) – (c*b)|
    • (Note: The determinant is the absolute value of the difference between a*d and c*b.)
• Example:
  • Matrix B:
    • [5, 4 | 9, 8]
  • Determinant of matrix B:
    • Det (B) = |(5 * 8) – (9 * 4)|
      • = |40 – 36|
      • = 4

It’s helpful to find out if the determinant of a matrix equals 0 as this will tell you if there is or isn’t an inverse to the matrix. To be honest, I don’t really understand why this is the case but I know that it has something to do with the fact that if the determinant equals 0 then the matrix transformation will scale the vector, 2×2 matrix, etc. to 0 which cannot be scaled back with an inverse operation. Here are two pages from my notes that explain:

On Friday I worked through a section titled Solving Linear Systems with Matrices. This section had me use the matrix inverse formula which I couldn’t remember and needed to review. Here’s the formula following by 2 example questions:

Question 1

Question 2

I actually got the second question wrong by adding the values in the second and third row of the vector in the final step incorrectly leading me to say that y = –17 and z = –15. I amended my note to make the solution correct by changing the (–) to a (+) at the bottom of the page.

I ended up passing the Matrices unit test on my first attempt and didn’t find it too difficult. And as I said at the beginning, I then found the videos and exercises in Probability and Combinatorics very easy and, to be honest, don’t think they’re worth going through here.

I’m hoping to get through the unit test of Probability and Combinatorics (1,240/1,400 M.P.) on Monday and move on to Calculus 2 by Wednesday. Right now I’m 61% of the way through Calc.2 having finished 6,420/10,500 M.P. from the course. I can’t remember how long it takes me on average to get through ~4k worth of M.P. but I’m hoping I can manage to do it in 8 weeks. If I can and can also manage to pass the course challenge in that amount of time, I’d be finished the course on September 6^th which is the first day of school and would be exactly 3 years since I first started KA. That would be SO satisfying if I can pull it off so I’m really hoping I can!