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.

Since $A$ maps $(1, 0, 0)$ to $a_1$, $(0, 1, 0)$ to $a_2$ and $(0, 0, 1)$ to $a_3$, $(1, 1, 0)$ and $(0, 1, 1)$ are both solutions to $Ax = b$. Linear transformations give only one, zero or infinite solutions, thus there are infinite solutions. Alternatively, $m(1, 1, 0) + n(0, 1, 1)$ is a solution for any $m + n = 1$.
