This chapter contains some important linear algebra as distinguished from that which is normally presented in undergraduate courses consisting mainly of uninteresting things you can do with row operations.
The notation, ℂn refers to the collection of ordered lists of n complex numbers. Since every real number is also a complex number, this simply generalizes the usual notion of ℝn, the collection of all ordered lists of n real numbers. In order to avoid worrying about whether it is real or complex numbers which are being referred to, the symbol F will be used. If it is not clear, always pick ℂ.
Definition 4.0.1 Define
it is conventional to denote
for t in the ith slot is called the ith coordinate axis. The point 0 ≡
The geometric significance of ℝn for n ≤ 3 has been encountered already in calculus or in precalculus. Here is a short review. First consider the case when n = 1. Then from the definition, ℝ1 = ℝ. Recall that ℝ is identified with the points of a line. Look at the number line again. Observe that this amounts to identifying a point on this line with a real number. In other words a real number determines where you are on this line. Now suppose n = 2 and consider two lines which intersect each other at right angles as shown in the following picture.
Notice how you can identify a point shown in the plane with the ordered pair,
You can’t stop here and say that you are only interested in n ≤ 3. What if you were interested in the motion of two objects? You would need three coordinates to describe where the first object is and you would need another three coordinates to describe where the other object is located. Therefore, you would need to be considering ℝ6. If the two objects moved around, you would need a time coordinate as well. As another example, consider a hot object which is cooling and suppose you want the temperature of this object. How many coordinates would be needed? You would need one for the temperature, three for the position of the point in the object and one more for the time. Thus you would need to be considering ℝ5. Many other examples can be given. Sometimes n is very large. This is often the case in applications to business when they are trying to maximize profit subject to constraints. It also occurs in numerical analysis when people try to solve hard problems on a computer.
There are other ways to identify points in space with three numbers but the one presented is the most basic. In this case, the coordinates are known as Cartesian coordinates after Descartes1 who invented this idea in the first half of the seventeenth century. I will often not bother to draw a distinction between the point in n dimensional space and its Cartesian coordinates.
The geometric significance of ℂn for n > 1 is not available because each copy of ℂ corresponds to the plane or ℝ2.