Notes preparatory to "rank"
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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