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A linear vector space is very democratic (or maybe "fair, open, and non-hierarchical" is better): any "competent" set of vectors can be a basis.
More specifically, for a n-dimensional vector space, any set of n independent vectors is a basis.
Here is an explanation without equations, something fun to think about on your next walk or to help you fall asleep.
Free Energy · Independent vectors and bases: a fun mental exercise for your next walkIn linear algebra, we learn that we can choose any set of \(n\) independent vectors from a \(n\)-dimensional vectors space \(V\), and that set will be a basis for \(V\). This is non-intuitive for me, and every once in a while, I realize that I’ve forgotten why this is so, and then have to re-learn it. So here is a quick-and-informal explanation with no equations that is hopefully easier to remember, and maybe it’s also a fun exercise to see if you can recreate this while on a walk. It largely follows the explanation from Tom M. Apostol’s Calculus Volume II (which did have many equations).