August 9, 2017

# Dimension is the minimum number of coordinates to specify a point in it The dimension of a space is the minimum number of coordinates needed to specify any point within it [1, 2].

1. Traditional two and three dimensional spaces;
2. Example dimensions for e-commerce services;
3. Vector, basis, and cardinality;
4. The dataset.

## 1. Traditional two and three dimensional spaces

For instance, any point on a sheet of paper could be specified in terms of its coordinates along each side of the sheet. Here the dimension would be two. An instance of space with three dimensions would be a geospatial environment in which height is added. There would be four dimensions if we add time.

## 2. Example dimensions for e-commerce services

In e-commerce services, some examples of dimensions could include the following:

• the amount of the order;
• how often the client has visited the website;
• how long it took the client to make the actual payment;
• whether the client ordered a t-shirt; or
• whether the client is a repeat customer.

## 3. Vector, basis, and cardinality

In mathematics, each dimension represents a vector ; together, the set of vectors represents the basis of a vector space. For e-commerce services, we gave five possible dimensions, but there could be tens of thousands of dimensions or even millions!

The number of dimensions is called the cardinality  of the basis or, more commonly, the dimension of the problem. Moreover, the process of converting the information collected, e.g., a client’s purchases online, to a vector space is called a projection  or a map to a vector space representation.

## 4. The dataset

The set of clients who made purchases on the e-commerce service is called the data set . Each client is called a data point . In mathematics, a data set is represented as a data matrix or just a matrix , i.e., a rectangular array composed of numbers where the number of rows corresponds to the number of data points and the number of columns is the dimension of the vector space.

## Reference

1. Wikipedia. Dependent and independent variables.; 2017.
2. [Wikipedia. Vector space.] https://en.wikipedia.org/wiki/Dimension_(vector_space)); 2017.
3. Joseph Grifone. Algèbre Linéaire. Cépaduès-Éditions, 1994.
4. [Wikipedia. Projection.](https://en.wikipedia.org/wiki/Projection_(linear_ algebra%29); 2017.
5. Wikipedia. Data set; 2017.
6. Wikipedia. Data points; 2017.
7. Alksentrs at the English language Wikipedia [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)]