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.
Addendum, following the answers and the questions below: I am working on a data structure where points are taken from $\left\{ 0,1\right\}^{m}$ where m can potentially go to $\infty$. At a given moment, I only have points taken from $\left\{ 0,1\right\}^{n}$ where n < m.
Then, if I want to add an extra dimension, I have m-n possibility to do so - which may be a lot. Instead of doing this arbitrarily, I would like to have a kind of principled way or at least a mathematical understanding of such an operation (i.e. that takes as input N points from E(n) and returns a basis for E(n+1).
Thanks again!
Let $E(n)$ be a given vector space over the field $K$ and define $E(n+1)$ by $E(n+1)=E(n)\oplus K$ where $K$ is seen as a $1$-dimensional vector space over itself and $\oplus$ is the direct sum of vector spaces. Given that $E(n)$ is an $n$-dimensional vector space over $K$, we see that $E(n+1)$ is an $n+1$-dimensional vector space over $K$ from basic linear algebra.
If $B$ is a basis for $E(n)$ then $B'=\{(v,0_K)\mid v\in B\}\cup\{(0_{E(n)},1_K)\}$ is a basis for $E(n+1)$.
Any other $n+1$-dimensional vector space over $K$ with an embedded $E(n)$ is naturally isomorphic to this vector space via an isomorphism which restricts to the identity on the embedded $E(n)$ subspaces.