What is the number $N^d_k$ of real-valued parameters that are needed to specify a k-dimensional subspace of $mathbb{R}^d$? And how can these parameters be interpreted?
I know: $N^d_1 = N^d_{n-1} = d - 1 = binom{d}{1} - 1$.
The parameters can be interpreted as the d components of a vector spanning the 1-dimensional subspace minus its (arbitrary) length.
I know: $N_1^3 = N^3_2 = 2 = binom{3}{2} - 1$.
The parameters can be interpreted as two angles or as the three components of a normal vector of the 2-dimensional subspace minus its (arbitrary) length.
I know: $N^d_2 = N^d_{d-2} = binom{d}{2} - 1$
I believe this, because a d-dimensional rotation has $binom{d}{2}$ degrees of freedom, one for the rotation angle, the remaining $binom{d}{2} -1$ ones for the (d-2)-dimensional (hyper)plane of rotation which also defines a 2-dimensional hyperplane as its orthogonal complement.
Question: How do I know that $binom{d}{2}$ is the number of degrees of freedom of a d-dimensional rotation?
How can these $binom{d}{2}$ parameters of a rotation or the $binom{d}{2} - 1$ parameters of a 2-dimensional hyperplane be interpreted (maybe even intuitively)?
I guess that $N^d_k$, the number of parameters that are needed to specify a k-dimensional subspace of $mathbb{R}^d$, is given by $binom{d}{k} -1$. How can this be shown? Only formally by mathematical induction or more directly, using e.g. the observation, that there are $binom{d}{k}$ k-dimensional subspaces of $mathbb{R}^d$ spanned by k of d elements of an orthonormal basis of $mathbb{R}^d$?
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