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This module will explore the basics of vector representations; explaining what vectors are, how they represent data in information retrieval and how similarities can be computed using mathematical methods.
Vectors are fundamental mathematical objects used extensively in computer science and data science. A vector, $\textbf{v}$, is typically described by a list of numbers [1]:
$$ ⁍. $$
Note, they do not have to include only integers but any other number too, including decimals, fractions, etc.:
$$ ⁍. $$
More generally, we write a vector, $\textbf{v}$, in the form:
$$ \textbf{v} = (v_1, v_2, \dots, v_n), $$
where $n$ represents the number of dimensions of the vector (also the same as the number of elements in that vector).
For example, a vector comprised of two numbers is known as a two-dimensional vector. Take $\textbf{v} = (1, 1)$ as an example. This can be represented on a two-dimensional $(x, y)$ grid as follows:
Figure 1a: Representation of a two-dimensional vector.
where $\textbf{v} = (x = 1, y = 1)$.
We can also extend vectors to three-dimensions as seen below. They can be extended to even higher dimensions but that gets a little tricky to visualise!