Can any collection of numbers be considered as a vector?

Question

Let's assume I am considering a collection of numbers - $A=[1,7,6,3,4,7]$. So can $A$ be considered as a vector. And if yes then will will any collection of numbers could be considered as a vector ??

Answer 1

The word "vector" is used with slightly different meanings. In computer science it is sometimes synonymous to "list". So it that case, yes!

But most often it means "element of a vector space". So if by "number" you mean "real number", ie "element of $\mathbb{R}$", then yes, any such tuple can bee seen as a vector, because it is an element of $\mathbb{R}^n$, where $n$ (finite or infinite) is the length of the tuple. $\mathbb{R}^n$ can naturally be given the structure of a vector space (it is a product space of $\mathbb{R}$). So in this case, again yes!

Answer 2

Calling something a vector strongly implies it's membership in a vector space, which is a full algebraic structure as well as merely a combination of numbers. That's a lot of additional information which should be present to call something a vector.

However given two sets with $a \in A$ and $b \in B$ then I can define a tuple $(a,b)$ to be an element of the cartesian product $A \times B$. This way we can build up arbitrary collections of ordered lists of set elements. If I had a product of $n$ sets then I'd call that an n-tuple.

Also, if we restrict ourselves to the idea of number we can consider module over a ring as well. Since the integers form a ring but not a field we can still analogously use some of the techniques from linear algebra in this more general setting.