I took a month long break, so I thought I would make a quick forum post. Vectors can, in some sense, be described as a tuple of real numbers. Don't think this as "all vectors are tuples of real numbers," but instead "one way to look at vectors is tuples of real numbers." A vector dis still an element of a Vector Space described in my post "The Real Definition of a Vector Space," but now we will describe something called a basis. You've probably heard of these, but now we will make this more general. We say a subset *B ⊂ V*, for some Vector Space *(V, +, ·), *is a basis if *∀ v ∈ V ∃*finite *{e _{i}}_{i=1}^{n} ⊂ B ∧ {α_{i}}_{i=1}^{n} ⊂ ℝ* such that

*α*.

_{1}e_{1}+ ... + α_{n}= v

What this means is that there is a set which has a span (all linear combinations) that covers the entire Vector Space, with infinitely many zeros for each vector. From this, if *B* is finite, then the dimension of the Vector Space is the number of vectors in *B*, say *n*. Then, given this finite basis, we can define the components of a vector *v**:*

Given the constants *{α _{i}}_{i=1}^{n}* such that

*Σ α*. Then, we can associate

_{i}e_{i}= v*v*with the tuple

*(α*. Thus, we have our definition of the components.

_{1}, α_{2}, ... , α_{n})That one guy that's always on