I underestimated how difficult Precalculus would be. I assumed it would mostly be review and I would more-or-less breeze through it but this week I learned a number of brand new and difficult concepts that I’d never been shown before. I was a bit under 20% of the way through Matrices when the week began and finished the week 71% of the way through it. I worked for ~1.5 to 2 hours each day so I’m happy with the effort I put in but, nonetheless, I’m still disappointed it’s taking so long. That seems to be a running theme with my recent blog post. 😔

Me week started off working on matrix equations, specifically adding and subtracting matrices, and multiplying a matrix by a scaler. Each of these operations turned out to be fairly simple. Here are two photos that breakdown each:

• Matrix Addition/Subtraction
  • You can add and subtract matrices to and from each other as long as they have the same dimensions.
  • In the same way that you can manipulate linear equations using addition/subtraction to solve for variables:
    • Ex. 12 + a = 4
      • 12 + a – 12 = 4 – 12
      • a = –8
  • You can do the same thing with matrices as long as you’re adding/subtracting the corresponding elements from each matrix.
• Scalar Multiplication
  • Just like addition and subtraction, you can multiply (and therefore divide) a matrix with a scalar the same way you would a linear equation:
    • Ex. 4 * a = 12
      • (4 * a)/4 = 12/4
      • a = 3
  • You must remember to multiply every element inside the matrix by the scalar. (Duh.)

After easily getting through the addition/subtraction/scalar multiplication section my confidence was abruptly crushed as I was introduced to matrix multiplication which is when you multiply one matrix by another. This doesn’t require any exceedingly difficult math but does require ~10, 000 steps per question which makes it easy to make a mistake somewhere along the way. Before I explain the process of matrix multiplication, you must first understand the difference between when matrix multiplication is defined vs when it’s undefined (read: when you’re “allowed” to multiply matrices together vs when you’re “not allowed” to multiply them together):

To reiterate what the first photo says, matrix multiplication is only possible when the first matrix has the same number of columns as the number of rows in the second matrix that it’s being multiplied by. If that’s the case, the equation is considered “defined” meaning it’s possible to dot he multiplication. If the number of rows in the second matrix doesn’t match the number of columns in the first matrix than the equation is considered “undefined” and it’s impossible to multiply them together. (I should note that Sal said a few times that this is based off standard matrix multiplication conventions making it sound as if there may actually be a way to multiply “undefined” matrices together but that it’s not common practice.) As demonstrated in the second photo, this means that unlike linear multiplication, order matters in matrix multiplication since the columns in the first matrix must match the rows in the second matrix.

Once you know the difference between defined and undefined matrices you can learn the process of multiplying defined matrices together:

I find it very difficult to precisely put into words the process of calculating each of the elements in the product of two matrices. Essentially, you multiply the elements in the row of the first matrix by the elements in the column of the second matrix for each element in the product. (SO HARD TO EXPLAIN.) Moving on…

On Thursday I learned about the Identity matrix which is the equivalent of the value 1 when multiplying linear equations:

• As far as I know, the dimensions of this matrix is always square.
• Since it’s always a square, it’s denoted with I_n since n implies n*n.
• An identity matrix always consists of 1’s and 0’s with 1 being used on a diagonal from the top left corner of the matrix to the bottom right corner of the matrix and 0 being used for the rest of the elements.
• I don’t know why this matrix is useful.

I then was introduced to the most difficult topic of the entire week, matrix transformation. I’m not going to begin to try and explain in detail how it works since I have such a weak understanding of it, but here’s an example of a question I went through:

In the question, I had to take the purple pentagon and transform it by multiplying it by the vector R. I first had to determine the coordinates of each of the five points of the pentagon and write them in matrix form (which I did underneath where it says “Pentagon”). I then used matrix multiplication to multiply vector R by the pentagon’s matrix. (I simplified it in my notes, but that calculation was done in the bottom half of the photo.) The resulting product (top right corner) was the coordinates of the pentagon’s transformation, i.e. the green pentagon in the screen shot.

As of now I don’t have the slightest intuitive understanding of how matrix multiplication works. I was shown a number of videos that demonstrated how shapes expand, shrink, and rotate using matrix multiplication and found it all very confusing. It’s frustrating but I’m confident that if I keep grinding away at it I’ll eventually figure it all out.

 I then learned about the determinants of matrices, and specifically 2×2 matrices:

Understanding how to calculate the determinant of a matrix is necessary to know in order to calculate the inverse of a matrix:

In the same way that if you multiply an integer by it’s inverse you get 1 (ex. 4 * (1/4) = 1), multiplying a matrix by it’s inverse results in the identity matrix, i.e. the matrix equivalent of 1. Here’s the formula to calculate the inverse of a matrix and an example of how it’s used:

If you multiply matrix B in the photo above by it’s inverse, B^(-1), you are left with the identity matrix:

So ya, matrices seem pretty tough to me right now. I had a moment this week where I was so discouraged I genuinely thought for a minute that I wasn’t smart enough for this type of math, but then I did a 180 and thought, “fuck that, I’m going to keep grinding and will get this eventually.” (lol.) I reminded myself that at times I felt the same way about trigonometry and I eventually figured it all out.

I’m not going to get through Precalculus by the end of this week which is fine. My goal this week is to get through the remainder of Matrices (1200/1700 M.P.) which I’m fairly confident I’ll be able to do. One thing I was happy to learn this week was that the courses Calculus 1, Calculus 2, and Linear Equations are apparently college math courses. I think it’s cool that, once I get through them all, I’ll understand college level math. More like COOL-lege level math if you ask me! 😁