I am stuck on the following homework problem: Show that if S=S(m,n,r) represents the space of m by n matrices with rank less than or equal to r (naturally isomorphic to an affine subvariety of $\mathbb{A}^{mn}$ of course) over some arbitrary field k, then if r < min(m,n), the zero matrix is a singular point of S (in the sense of tangent spaces).
Now for an affine subvariety determined by the zeros of some $f_i$, I know the definition of a tangent space in terms of the directional derivatives etc., and I guess I want to find that, but I am confused about what the $f_i$ I would be taking directional derivatives of are: our matrices have rank $\leq$ r iff every r+1 by r+1 minor of the matrix has zero determinant, so these are the "equations" for which our matrices form the vanishing set: however, there are many many such equations and clearly it is not feasible to work with all of them at once. If our r is strictly less than the minimum of the row/col size then I think this implies that there actually is such a minor for which the determinant is 0, but then to show that a point is singular I would need to show that the dimension of the tangent space does not equal the dimension of S, wouldn't I? This surely implies I know the dimension of S, which is entirely non-obvious to me. Or maybe it is just that in the case of the zero matrix, it is easy to show the dimension of the tangent space is not equal.
Otherwise, is there a better (equivalent) definition of 'singular point' I could work with here? Preferably one where the equivalence of definition is reasonably simple to establish, since it seems a bit pointless having a proof of equivalent definition longer than the actual solution. Where am I going wrong? I am only a fourth year mathematician and algebraic geometry isn't my strong point so please try not to go overboard! -Peter