In the euclidean space ${\mathbb R}^3$, I can define a plane by three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$, using six reals. Of course I can give an equation $ax+by+cz=d$, using only 4 reals.
But I can define a plane using a normalize equation ($a^2+b^2+c^2=1$), so I can rewrite it as $\cos\theta.x+\sin\theta\cos\phi.y+\sin\theta\sin\phi.z=d$, and I only need 3 reals.
I was wondering how many reals I need to define "canonically" a flat subspace of dimension $d$ into ${\mathbb R}^n$. If $d=0$, the flat is a point and the obvious answer is $n$. But what happens for other value ? How can I define canonical equations or an orthogonal base from such a definition ?
Of course, as ${\mathbb R}^n$ has the same cardinal as ${\mathbb R}$, you can answer there is a bijection between $\mathbb R$ and the set of flat subspaces but this is not my question as I want something constructive (and in some way continuous and naive).
So I want some function $f$ from $\mathbb R^k$ into the set of flat subspace of dimension $d$ (of $\mathbb R^n$). And I want $f$ to be surjective, continuous and $k$ minimum. So there must be some function $\phi(d,n)=k$, that gives the smallest such $k$ for given $d,n$.