Suppose I have an NxM matrix, where N > M. I know the rank of the matrix is M, which means there's a set of M-vectors forming the basis of the matrix. There's a word for this, and I can probably make mathematica do it easily, but I can't remember what it's called, so I can't google search it, and I don't have a linear algebra book. Help! PS---I hope this doesn't destroy everyone's confidence in me as a physicist!

If you can find M vectors forming a basis for the column space (or the range) of the matrix, then the rank is M, provided that . But there are matrices (e.g., one with two identical columns) that have rank less than M. The word you are looking for might be: column space, range, dimension of the range, linearly independent columns

This is related to the rank-nullity theorem. If the dimension of the kernel of your matrix A is m and it maps into \(\mathbb{R}^{M}\) then you have that the rank of A is M-m. http://en.wikipedia.org/wiki/Rank_of_a_matrix#Computation . While it's an interesting ( use the word in a vague way) to program your own Gaussian elimination routines (damn you 2nd year course work!!) in C, Mathematica already has all that programmed in and I think just something like Rank[A] will give you an integer back less than or equal to your M.

The name of it is just simply "column rank"; It is called the column rank of the matrix because there are M linearly independent columns. So, if you have M pivots in an NxM matrix (after reducing it to Row Echelon Form), it is said to be the Column rank of the matrix. Likewise, if you have N pivots (a pivot in every row), then N is the row rank of a matrix.