Definition 1.2.1.
In linear algebra and many other brances of mathematics, ordinary numbers (integers, rational numbers or real numbers) are called scalars.
Section 1.2 Euclidean Space
Subsection 1.2.1 Definitions
Euclidean or cartesian space is the environment for this course. The cartesian plane is familiar to you, but I’m going to give formal definitions regardless. This is part of the process of learning the formal language of mathematics: of taking previously known mathematical ideas and redefining them in more formal, logical ways. We start that process with geometric space and the vectors therein.
Definition 1.2.2.
A vector is a finite ordered list of scalars. Vectors can be written either as columns or rows. If a vector is a list of the numbers \(4\text{,}\) \(15\text{,}\) \(\pi\) and \(-e\) (in that order), I write the vector in one of two ways:
\begin{align*}
\amp \begin{pmatrix} 4 \\ 15 \\ \pi \\ -e \end{pmatrix}
\amp \text{or} \amp \amp (4, 15, \pi, -e) \amp
\end{align*}
In this course, I will exclusively use column vectors. In many other places, row vectors are common. I find it best to choose one of the two for the whole course instead of switching back and forth. For reason that I’ll talk about in future weeks, column vectors are a good choice for this course.
Definition 1.2.3.
Let \(n\) be a positive integer. Real Euclidean Space or Cartesian Space, written \(\RR^n\text{,}\) is the set of all vectors of length \(n\) with real number entries. I write an arbitrary vector with coefficients \(x_i\) as follows.
\begin{equation*}
\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}
\end{equation*}
Since there are \(n\) coefficients in its vectors, the space \(\RR^n\) has dimension \(n\text{.}\)
Two dimensional space, \(\RR^2\text{,}\) and possibly three dimensional space, \(\RR^3\text{,}\) should be familiar. Notice, though, that my new formal definitions work in any dimension. This is the power of mathematical abstraction and formalization. Even though we live in three dimensions and can’t visualize higher dimension, there are many reasons (including very practical reasons) to consider higher dimensions. Therefore, my definitions are flexible enough to consider four dimension space, or six dimension vectors, or whatever else I need.
Definition 1.2.4.
The scalars \(x_i\) in a vector are called the entries, coordinates or components of that vector. Specifically, \(x_1\) is the first coordinate, \(x_2\) is the second coordinate, and so on. For \(\RR^2\text{,}\) \(\RR^3\) and \(\RR^4\text{,}\) I use the letters \(w,x,y,z\) instead of \(x_i\text{.}\)
\begin{align*}
\begin{pmatrix} x \\ y \end{pmatrix} \amp \in \RR^2 \amp
\begin{pmatrix} x \\ y\\ z \end{pmatrix} \amp \in \RR^3
\amp \begin{pmatrix} w \\ x\\ y\\ z \end{pmatrix} \amp
\in \RR^4
\end{align*}
Definition 1.2.5.
In any \(\RR^n\text{,}\) the origin is the unique point given by a vector of zeros. It is also called the zero vector. It is considered the centre point of Euclidean space.
\begin{equation*}
\begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix}
\end{equation*}
This is geometry, so I want to actully see it. Euclidean space is usually visualized by drawing axes, one in each independent perpendicular direction. In this visualization, the vector \(\begin{pmatrix} a \\ b \end{pmatrix}\) corresponds to the unique point I get moving \(a\) units in the direction of the \(x\) axis and \(b\) units in the direction of the \(y\) axis. Figure 1.2.6 shows the location of several points in \(\RR^2\text{.}\)
As with \(\RR^2\text{,}\) the point \(\begin{pmatrix} a \\ b \\ c
\end{pmatrix} \in \RR^3\) is the unique point I find by moving \(a\) units in the \(x\) direction, \(b\) units in the \(y\) direction and \(c\) units in the \(z\) direction. When visualizing \(\RR^2\text{,}\) the convention is to draw the \(x\) axis horizontally, with a positive direction to the right, and the \(y\) axis vertically, with a positive direction upwards. For \(\RR^3\text{,}\) the \(x\) and \(y\) axes form a flat plane, and the \(z\) axis extends vertically from that plane. Figure 1.2.7 shows some points in \(\RR^3\)
Definition 1.2.8.
Note that I needed to choose directions for the axes in both \(\RR^2\) and \(\RR^3\text{.}\) In a visualization of Euclidean space, there are no predetermined directions for axes. I have to make a choice. This choice of axis directions in the visualization of \(\RR^n\) is called orientation.
While I can visualize \(\RR^2\) and \(\RR^3\) relatively easily and efficiently, I can’t visualize any higher \(\RR^n\text{.}\) However, this doesn’t prevent me from working in higher dimensions. I need to rely on the algebraic descriptions of vectors instead of the drawings and visualizations of \(\RR^2\) and \(\RR^3\text{.}\) This is where the algebra/geometry connection becomes truly remarkable: if I build the algebra to describe two and three dimensional geometry, I can use that algebra to describe four, five and higher dimensional geometry as well. Even though I can’t see these spaces, I can understand them and work with them. It’s amazing.
In the visualizations of \(\RR^2\) and \(\RR^3\text{,}\) I see the different axes as fundamentally different perpendicular directions. I can think of \(\RR^2\) as the space with two independent directions and \(\RR^3\) as the space with three independent directions. Similarly, \(\RR^4\) is the space with four perpendicular, independent directions, even though it is impossible to visualize such a thing. Likewise, \(\RR^n\) is the space with \(n\) independent directions.
Subsection 1.2.2 Points or Directions?
I can think of an element of \(\RR^2\text{,}\) say \(\begin{pmatrix} 1 \\ 4 \end{pmatrix}\) , as both the point located at \(\begin{pmatrix} 1 \\ 4 \end{pmatrix}\) and the vector drawn from the origin to the point \(\begin{pmatrix} 1
\\ 4 \end{pmatrix}\) , as shown in Figure 1.2.9. Though these two ideas are distinct, I will frequently switch between them, often without explicitly saying so. Part of becoming proficient in vector geometry is becoming accustomed to the switch between the perspectives of points and directions.
Subsection 1.2.3 Local Direction Vectors
I’ve already spoken about the distinction between elements of \(\RR^n\) as points and vectors. There is another important subtlety that shows up all throughout vector geometry. In addition to thinking of vectors as directions starting at the origin, I can think of them as directions starting anywhere in \(\RR^n\text{.}\) I call these local direction vectors.
Figure 1.2.10 shows local direction vectors starting at the point \(\begin{pmatrix} 2 \\
2 \end{pmatrix} \in \RR^2\text{.}\) The two local vectors \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) are relative to the point \(\begin{pmatrix} 2 \\ 2 \end{pmatrix} \in \RR^2\) as if that were their origin.
Using vectors to define local directions is a useful tool. A standard example is a camera in a three-dimensional virtual environment. First, I need to know the location of the camera, which is an ordinary vector starting from the origin. Second, I need to know what direction the camera is pointing, which is a local direction vector which treats the camera location as the current origin.
One of the most difficult things about learning vector geometry is becoming accustomed to local direction vectors. I won’t always carefully distinguish between vectors at the origin and local direction vectors; often, the difference is implied, and it is up to the reader/student to figure out how the vectors are being used.
