If I have a set of A-conjugate vectors ( given two vectors a and b you
can say they are A-conjugate if aAb=0 with A a NxN matrix) can I imply
that they are all linearly independent?

The simplest way for this to fail would be:

xAx = 0

for some nonzero vector x.  For example:

A = +1  0
     0 -1

and x = (1,1).

Clearly some special properties of A are needed to make the statement
true.  Note that if A = I, then saying a set of nonzero vectors are
A-conjugate amounts to saying they are orthogonal (and hence linear
independence does follow).

Given your knowledge of the conjugate gradient algorithm, would you
care to guess what conditions on A are sufficient for your purpose?

regards, mathtalk-ga

Technically we need to take x' to mean transpose of x, so that if x is
a row vector, then xAx' = 0 is defined to be a scalar result (of a
matrix product involving row x, matrix A, and column x').

-- mathtalk-ga