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How to build a basis for a vector space E(n+1) from a set of points given in E(n) (a vector space of rank n).

I'm interested in how (and if) one can build a new dimension from a set of given dimensions. Specifically, if we are given a vector space E(n) of rank n, and a sample S of elements of E(n) (let us say, S arbitrarily big):

Can we build a vector basis for some E(n+1) of rank n+1?

I'm also interested in keywords or themes that study this kind of questions in maths (if any).

I've been looking up for Lie brackets, unstable operations in vector fields, and words such as involutivity and extension algebras.

Thank you.