Meaning of $x_i$ in a vector $\mathbf x=(x_1,\dots,x_n)$?

Question

That is the meaning of $x_i\in\mathbb R$ in the following? I mean, what is the index $i$?

A vector is $$\mathbf x=(x_1,\dots, x_n), \quad x_i\in\mathbb R $$

Answer 1

The index is just used to specify that $x_{i} \in \mathbb{R}$, where $1 \leq i \leq n$. In other words, if $n$ was 3, then $x_1$, $x_2$, and $x_3$ would all be real numbers.

Answer 2

It means that $x_i \in \mathbb{R}$ for all $1 \leq i \leq n$, i.e. $x_1 \in \mathbb{R}, x_2 \in \mathbb{R},$ etc.

Because the range of $i$ is obvious from writing $x$ right in front, it was left out.

Answer 3

The $i$ stands in for “any of the indices.” It’s really just a shorthand. The full expression should be something like $$\bigcup_{i=1}^n\{i\}\subset\Bbb R$$

The origin is somewhere in “index notation” of general relativity, wherein statements rely almost entirely on indices and physicists drop summation symbols by convention. For example,

$${\bf w}\cdot{\bf z} = \sum_{i}w_iz_i = w_iz_i$$

where ${\bf w} = \pmatrix{w_1 & \cdots & w_n}^{\sf T}$ and ${\bf z} = \pmatrix{z_1 & \cdots & z_n}^{\sf T}$.

Answer 4

Intuitively, the entries in a vector may be thought of as coordinates in a system. They are part of a location, and all the coordinates together constitute the direction of the vector (location in a system).

Each coordinate, $x_i$, represents some important aspect of the system.