Notes preparatory to "rank" ===== =========== == ==== [If at any time you wish to exit this page, please click your browser's [Back] button.] This note contemplates a k by n matrix A . Whenever a k-deep column matrix b is given, we may inquire whether there are any n-deep columns x for which the matrix equation (#) A x = b is valid. This equation is just the matrix counterpart of a certain corresponding system of simultaneous linear equations whose coefficient matrix is A , and whose augmented matrix M has the form M = [ A | b ] . What do we really know about A if we know that, (1) for each conceivable k-deep column matrix b , equation (#) always has one or more solution(s) x ? Certainly we know that the fully reduced row echelon form (frref) of the augmented matrix M never has any "embarrassing" row starting with n zeros but ending with a non-zero entry in the last column, and this regardless what the original k-deep column matrix b . But this means exactly that (2) the frref of A itself has no row of zeros whatsoever. (From this condition on A , in turn it is easy to deduce (1).) (Of course (1) or (2) can happen ONLY if there are at least as many variables as equations, i.e., k