Linear Combinations and solutions Let A be a 5 x 3 matrix. If $$b = a_1 + a_2 = a_2 + a_3$$
then what can you conclude about the number of solutions of the linear system Ax = b? Explain.
I'm not sure about this question. All I know is that if b can be written as a combination of column vectors a, then the linear system is consistent. I am not sure what this says about the number of solutions, however.

Yeah, old question, but I came across it so someone else might, as well.  More clearly, since b is a combination of vectors a1, a2, and a3 (either [1,1,0] or [0,1,1]), the system MUST be consistent and have at least one solution.  Because the system is overdetermined (5x3), it could be exactly one solution OR infinitely many solutions, depending on whether the system is linearly independent.
MH234's system is underdetermined (3x4) so one of the variables is necessarily free and provides for infinitely many solutions.
