The following question might be connected to Embed curves in the plane
Given a curve in $\mathbb{R}^n$ (smooth and so on), what is the smallest basis in which every point of the curve can be written as a linear combination of the basis vectors?
For example $g(t) = (t, t^2, t)$ is defined in $\mathbb{R}^3$, but lies completely in the subspace spanned by vectors $(1,0,1)$ and $(0,1,0)$.
I'm not experienced in topology, so I can't tell whether what I want is equivalent to what "embedding" normally means. My question though is, how to explicitly construct the smallest linear (sub)space that contains a given curve.
Or is it only possible for trivial cases, like the one I chose above?
EDIT: Maybe to put the question in more symbolic Notation
For a given differentiable function $g:\mathbb{R}^+ \rightarrow \mathbb{R}^n$, with $g(\mathbb{R}^+) \subset [0,1]^n$ we define $\mathcal{B}_g = \{v_i \in \mathbb{R}^n \,, \forall \, i = 1,\ldots,k\}$ to be a Basis iff
$(\forall \, t \in \mathbb{R}^+)(\exists \, \alpha_1,\ldots,\alpha_k \in \mathbb{R}) \, \, g(t) = \sum\limits_{i=1}^k \alpha_i v_i $
Question: How to construct the smallest Basis for a given curve $g$?