Textbook
In the coordinate model, a vector $x\in\mathbb R^d$ is an ordered collection of $d$ real coordinates, where integer $d\ge1$. The number $x_i$ is coordinate $i$. The examples use one-based indexing and column vectors.
For instance, $x=(2,4,6)^{\mathsf T}\in\mathbb R^3$. The symbol $\mathsf T$ transposes the displayed row into a column; it is not an exponent. Equal vectors have equal dimensions and equal corresponding coordinates. Order matters: $(2,4)^{\mathsf T}\ne(4,2)^{\mathsf T}$. Abstract linear algebra defines vectors through vector spaces; this article uses the concrete numerical-coordinate model.
A coordinate index must lie in its declared range: with one-based indexing, $1\le i\le d$. The selected $x_i$ is a number, whereas $x$ is the entire ordered collection; replacing one by the other changes the object’s type. Repeated coordinate values are allowed and do not reduce dimension. Equal values in two positions do not permit deleting one position without changing the vector itself.
Definitions and further reading: Mathematics for Machine Learning.
In plain language
A coordinate vector resembles a form with fixed fields. Exchanging height and age changes the meaning even when the collection of numbers stays the same. Dimension specifies the number of fields rather than their names or units. Two files with three columns need not describe the same features; unlabeled numbers can look convincing too.
Example
The original illustrative vector is $x=(2,4,6)^{\mathsf T}$. Its coordinates are selected in sequence; reading them does not change the vector.
Coordinate Vectors: Order and Dimension
Coordinate 1
The first index selects 2.
Step 1 of 3: Coordinate 1
Read all steps
Coordinate 1
The first index selects 2.
- State
- 2current46
Coordinate 2
The second position contains 4.
- State
- 24current6
Coordinate 3
The third position contains 6; there are three coordinates.
- State
- 246current
The scene’s vector stays $(2,4,6)^{\mathsf T}$ throughout: the selected position changes, not the source data. Selecting index 2 returns 4 rather than the first two values together. Index 4 is unavailable because only three coordinates exist. In zero-based code, the mathematical second coordinate would occupy position 1. That indexing convention requires an explicit translation, not a change to the underlying vector.
Related concepts: Transposition, Vector norms.