# 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.

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How is your question related to Sage ? Even mathematically, your question is not precise enough. For example, if E(n) is a vector space of dimension n over a field F, you can always create a new space $E(n+1) = E(n)\times F$ as a vector space of dimension n+1, but what is the meaning of this additional dimension ?