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Section 23.1 Motivation

Subsection 23.1.1 Complex \(n\)-dimensional vectors

When we first met vectors in Chapter 12, we conceived of them as displacement arrows in the plane or in space that describe how to move from an initial point to a terminal point. Attached to such a displacement arrow are the actual displacement amounts in each direction, and we recorded these displacement amounts as an ordered pair \(\uvec{v} = (\Delta x, \Delta y)\) (for a vector in the plane) or as an ordered triple \(\uvec{v} = (\Delta x, \Delta y, \Delta z)\) (for a vector in space). We immediately generalized this method of representing vectors by numbers, by extending to “higher-dimensional” vectors \(\uvec{v} = (v_1, v_2, \dotsc, v_n)\text{.}\)

But such vectors are not constrained to geometric contexts. Instead of representing changes in spatial variables as an object moves from some initial point to some terminal point, a vector can represent coordinated change in any collection of related variables, between some initial state and some terminal state. (For example, see Discovery 12.11.)

What if we have a collection of related complex variables for which we would like to represent change as ordered lists of numbers? Then our ordered lists need to contain complex numbers.

Example 23.1.1. Complex displacement vectors.

If an object “moves” from “point” \((z_1,z_2) = ( 1 + \ci, 3 - 4 \ci )\) to “point” \((z_1,z_2) = ( -\ci, 5 + 4 \ci )\text{,}\) we can calculate the change in each variable:

\begin{align*} \Delta z_1 \amp = (-\ci) - (1 + \ci) = -1 - 2 \ci\text{,} \amp \Delta z_2 \amp = (5 + 4 \ci) - (3 - 4 \ci) = 2 + 8 \ci\text{.} \end{align*}

Putting these together in an ordered pair, the “displacement” of the object can be represented as the complex vector

\begin{equation*} \uvec{v} = ( -1 - 2 \ci, 2 + 8 \ci) \text{.} \end{equation*}

Considering the above example, we can say that if we would like to use vectors to represent change in collections of related complex variables, then generalizing the concept of \(n\)-dimensional vector to allow complex components seems reasonable. We will explore such complex vectors further in the rest of this chapter.

Subsection 23.1.2 Complex abstract vectors

After exploring \(n\)-dimensional vectors with real components in some depth, we looked around and noticed that other types of mathematical objects (e.g. matrices, polynomials, functions) exhibit similar algebraic properties as \(n\)-dimensional vectors, and so in Chapter 16 we developed an abstract framework for describing the algebraic patterns common to each of these various contexts. This framework was built on the algebraic patterns satisfied by the two basic operations of \(n\)-dimensional vectors, vector addition and scalar-multiplication:

\begin{align*} (u_1,u_2,\dotsc,u_n) + (v_1,v_2,\dotsc,v_n) \amp = ( u_1 + v_1, u_2 + v_2, \dotsc, u_n + v_n )\text{,}\\ \\ k(v_1,v_2,\dotsc,v_n) \amp = ( k v_1, k v_2, \dotsc, k v_n )\text{.} \end{align*}

The very first rule for each of these operations is a “closure” rule: Axiom A 1 for vector addition and Axiom S 1 for scalar multiplication.

Adding two complex \(n\)-dimensional vectors will obviously result in another complex \(n\)-dimensional vector. And multiplying a complex \(n\)-dimensional vector by a scalar will obviously result in another complex \(n\)-dimensional vector. But that will be true even if we multiply by a complex scalar. So, in this chapter, when we collect the algebraic patterns of complex \(n\)-dimensional vectors into an abstract framework similar to the ten axioms in Definition 16.4.1, we will allow scalars to be complex numbers, instead of restricting them to be real numbers. However, we will find that this is the only modification to Definition 16.4.1 that we need to make in the complex context.