Lists

Before defining ${\mathbb{R}}^n$ and ${\mathbb{C}}^n$, we look at two important examples.

1. The set ${\mathbb{R}}^2$, which we can think of as a plane, is the set of all ordered pairs of real numbers:

$${\mathbb{R}}^2=\{(x,y):x,y\in \mathbb{R}\}$$

2. The set ${\mathbb{R}}^3$, which we can think of as an ordinary space, is the set of all ordered triples of real numbers:

$${\mathbb{R}}^3=\{(x,y,z):x,y,z\in \mathbb{R}\}$$

To generalize ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$ to higher dimensions, we discuss the concept of lists:

Definition: List, Length

• Suppose $n$ is a non-negative integer. A list of length $n$ is an ordered collection of $n$ elements (which could be numbers, other lists, or more abstract objects)
• Two lists are equal if and only if they have the same length and the same elements in the same order.
• A list of length $n$ is often called an $n$-tuple.

Lists are often written as elements separated by commas and surrounded by parentheses. Thus, a list of length two is an ordered pair that might be written as $(a,b)$. A list of length three is an ordered triple that might be written as $(x,y,z)$. A list of length $n$ might look like this:

$$(z_1,\dots ,z_n)$$

We sometimes use the word list without specifying its length. Remember, by definition each list has a finite length that is a nonnegative integer. Thus an object that looks like $(x_1,x_2,\dots )$, which might be said to have infinite length, is not a list.

A list of length 0 looks like this: $()$. We consider such an object to be a list so that some of our theorems will not have trivial exceptions.

Lists differ from finite sets in two ways: in lists, order matters and repetitions have meaning; in sets, order and repetitions are irrelevant.

• The lists $(3,5)$ and $(5,3)$ are not equal, but the sets $\{3,5\}$ and $\{5,3\}$ are.
• The lists $(4,4)$ and $(4,4,4)$ are not equal (they do not have the same length), but $\{4,4\}$ and $\{4,4,4\}$ both equal the set $\{4\}$.

Sometimes, when we consider lists of vectors, we do not use surrounding parentheses. For example, $(4,1,6),(9,5,7)$ is a list of length two of vectors in ${\mathbb{R}}^3$.