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?




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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)?




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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 $\mathbb{R}^d$?



How many parameters are needed to specify a k-dimensional subspace of R^d?
This answer is similar to what Ben said about $d \times k$-matrices, but maybe a little more visual. Instead of computing the dimension "as a whole", I'll just compute the dimension of subspaces neighboring a given one.


Take some fixed $k$-dimensional subspace $P$ with complementary space $P^\perp$ of dimension $n-k$. Any sufficiently nearby subspace $P'$ to $P$ looks like a graph of a linear function $A : P \to P^\perp$. And on the other hand, any such linear function defines a unique subspace. So you only need to count linear maps from $P$ to $P^\perp$, which are $k(n-k)$-dimensional.




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Thinking about the issue locally also helps avoid a mistaken dimension count like $n \choose k$. The problem with the $n \choose k$ count is that you are enumerating some random points on the Grassmannian, which doesn't tell you anything about the dimension. For example, take the case of 2-dimensional subspaces of $\mathbb{R}^3$. A subspace here is determined by its normal vector, so there is a bijection between 2-dimensional (oriented) subspaces and the unit sphere.
The $3 \choose 2$ planes in your count correspond to the north pole and two equatorial points $90^\circ$ apart. But you wouldn't conclude that the sphere is 3-dimensional just because it has three points!


