This was one of the most enjoyable weeks I’ve had working through KA so far. I finished the unit Vectors on Thursday and was able to get ~20% of the way through the following unit Matrices. Even though I wish I’d gotten further into Matrices, I was happy with my progress this week because 1) I still had to get through a lot of material in the remainder of Vectors and beginning of Matrices and 2) everything I learned I found to be challenging but not so difficult that I couldn’t eventually figure it out. There have been times that I’ve been stuck on certain concepts for weeks which is incredibly frustrating and this week seemed to be the right mix of challenging but manageable.
This week I learned about the three forms that vectors can be written in which are Component form, Unit Vector form, and Magnitude and Direction form:
I think of the first form shown in the photo, component form, as simple x-, y-coordinates except the value for each coordinate represents the “change-in-x” and “change-in-y” values of the vector and not necessarily the x-, y-coordinates where the end of the vector lands. This is important to remember when adding and subtracting vectors in this form.
Right now I don’t really understand the purpose of the second form shown in the photo, the unit vector form. As far as I know, this form uses i”-hat” and j”-hat” (as shown in the photo just below this) to denote the vector moving one unit in the positive x-axis and positive y-axis respectively:
I’m sure there’s a reason why it’s useful to add i”-hat” and j”-hat” to denote vectors but right now I don’t see the difference between writing a vector in Component form and Unit Form. Thinking about it, I suppose using unit vector form would be a bit more specific/precise since someone might be unsure whether a vector written in component form was indeed a vector or coordinates for a point a point on the x-, y-plane. Hopefully I’ll get more clarity on this when I get into calc.
The last vector form, magnitude and direction form, gives the length of the vector (a.k.a. the magnitude) and the angle relative to an x-, y-plane (a.k.a. the direction). You can easily figure out the Opposite and Adjacent side lengths (i.e. the “change-in-x” and “change-in-y” lengths of the vector) by using SOH CAH TOA where (Magnitude)Sin(θ) = Opposite and (Magnitude)Cos(θ) = Adjacent, since the magnitude is the hypotenuse. I learned that when adding two vectors in magnitude and direction form you can simply add the trig values together:
It took me two attempts to get through the unit test but, nonetheless, I feel very confident with my understanding of everything I learned about vectors. I moved on to the unit Matrices on Friday which I had no previous knowledge of before this week. For some reason I was a bit intimidated going into the unit but it turns out, from what I’ve learned so far, matrices aren’t that difficult of a concept to understand:
As described in the above photo, a matrix is an array of numbers displayed in a square or rectangle with nothing but square brackets on the outside of the set of numbers. Each number inside the dataset are known as Elements:
As you can see at the bottom of the above photo, you can denote a specific element within a matrix by first stating the variable used to denote the matrix followed by the specific row number and column number of the element written in subscript.
I then learned about Augmented Matrices which I still don’t understand the definition of at this point. It seems to simply be a matrix that represents a System of Equations, however I didn’t learn about a matrix representing anything other than this so I’m not sure what the purpose of labelling a matrix as Augmented is.
After that I learned about matrix Row Operations and how you can use them to solve a system of equations:
When I watched the first video where Sal used row operations to solve a system of equations and figure out the values of x and y, I didn’t know what he was doing initially and didn’t see it coming and my head exploded. In case the row operations photo was difficult to understand, the row operations I’ve learned so far are:
- Switch Rows
- You can move all the elements within a row up or down rows/switch rows with other rows.
- Denoted with R_1 <–> R_2
- (I think of the arrow as the word “replaces” as in “row 1 replaces row 2“)
- Multiply Row by Nonzero Constant
- You can multiply all elements within a row by the same number as long as it’s not zero. This is the same thing as when you multiply all values on both sides of an equation by the same number which keeps both sides of the equation equal.
- (#)R_1 –> R_1
- (“Any number multiplied by row 1 replaces row 1.”)
- Add Rows
- You can add all the elements from one row to all the elements from another row as long as you’re adding elements from the same columns.
- R_1 + R_2 –> R_2
- (“Row 1 plus row 2 replaces row 2.”)
The last thing I briefly learned about was adding and subtracting entire matrices to and from each other:
I only watched one video and didn’t go through any exercises on this so at this point I don’t know how/why adding and subtracting matrices will be useful but I’m sure it will be. The important thing to know seems to be that you can only add and subtract matrices that have the same dimensions.
This coming week my goal is simply to get through Matrices (320/1700 M.P.) since there’s still so much left to do in it. I think there’s a chance I could get through it all by Friday but it’s also equally possible that it may take me into next week to get throat done. Considering I have 3 units remaining after I get through Matrices (although none of them are very big), I’m starting to think it’s unlikely I’ll get through Precalculus by the end of February. As disappointed as I am by that, as I’ve said many times before, I’d much rather take the time I need to understand everything I’m going through as best as possible as opposed to rushing through the course to meet a deadline I’ve given myself. Still, after 76 weeks of studying math and wanting to learn calculus the whole time:
