Part 2: Vectors
The basis of linear algebra
As we move into February, my presentation is looming closer and closer. At this point, I have my outline roughly mapped out, and I am just researching some of the concepts that I am less comfortable with. Since my project is only supposed to be an introductory lesson, I am trying not to dig myself into a hole that is too deep, so today, I would like to go back a bit and take a look at the most basic concept in linear algebra: the vector. In reality, the vector is probably the most important concept in linear algebra as well, so it is going to be a crucial part of my presentation. For the rest of my presentation, I want to cover linear transformations and the matrix a little more closely, including matrix multiplication, determinants, null space, and then eigenvectors and eigenvalues. I also hope to convey the true meaning and ideas behind the mechanics of linear algebra, because that is where I believe the beauty lies in mathematics. Additionally, I see myself continuing this series after my project is completed at the end of February, as I have truly come to enjoy researching and writing about linear algebra.
Usually when I think of a vector, my first instinct is to think of physics. Not sure why, but that is just how it is. There are similarities and differences, but the linear algebra point of view of a vector is that it is just “an arrow,” centered at the origin, in space. Numerically, you can represent these vectors by their coordinates.
Here is a basic numerical and visual representation of a vector. In this form, we write the coordinates of the vector in a vertically rectangular bracket, with the x coordinate taking the top value and the y coordinate taking the bottom value. In three dimensions, we would add a z axis, and each vector would have three numbers in it, starting with the x coordinate first, the y coordinate second, and the z coordinate last. It is important to note that each pair of numbers within the bracket represents one and only one vector.
Vector addition, as it sounds, is just the sum of two vectors in space. Say we have two vectors. We’ll call them Vector A and Vector B. I have created a visual representation of the two, as well as their sum, below.
In words, we are taking Vector B, moving its origin point to the endpoint of Vector A, and then connecting the origin point to the new endpoint of Vector B. To me, it is almost like the steepness of the new vector is somewhat of an “average” of the steepness of the two original vectors. This is why we see A + B to be “inbetween” A and B (this may not make sense, but I am just reflecting on my initial thoughts on vector addition).
Numerically speaking, I guess you could say that vector addition is “easy.” The new x coordinate in the vector is simply the sum of the x displacement from the origin, and the new y coordinate in the vector is the sum of the y displacement from the origin. These new coordinates go into the new vector like usual, with the x coordinate on top and the y coordinate below. When we connect Vector A and Vector B, we move the x coordinate of Vector A plus the x coordinate of Vector B in the x direction (the sum), and we move the y coordinate of Vector A plus the y coordinate of Vector B in the y direction (the sum).
Numerically, we can see that we just add the numbers by row. On the top of the vector, we have 1 and 4, an on the bottom, we have 3 and 1. Therefore, the top, or x coordinate of our new vector, should be 1 + 4 = 5, and the bottom, or y coordinate of our new vector, should be 3 + 1 = 4.
Scaling a vector is simply multiplying a vector by a given constant. Note that this is NOT multiplying a vector by another vector. Visually speaking, when you scale a vector, let’s say by a factor of 3, the vector becomes 3 times as long. When you scale it by 1/2, you shrink the vector so that it is half as long as it was before. Furthermore, when negative numbers are involved, first flip the vector around and then perform the subsequent scaling.
The number we are multiplying by is called the scalar, and in the case above, the scalar is 2. Like vector addition, the mechanics of scaling are relatively simple. You simply need to multiply each of the x and y components of the vector by the given scalar.
With this research, I definitely took a step back. In reality, vectors are the basis of linear algebra, and I really should’ve learned about this before moving forward to something like matrices and linear systems. With this knowledge, some of the more difficult topics start to make sense, like linear transformations, matrices, matrix multiplication etc. Additionally, I also was able to keep in touch with the “visual learner” side of myself. Linear algebra became a lot easier once I started drawing pictures in my head. I now have part of the basics, which luckily weren’t too difficult to understand, down to delve deeper.