Suppose a vector $v_1$ is in the row space of some matrix $A$. Can we say anything about the column space? Gilbert Strang on 3:35 talks about this. Is vector $u_1=Av_1$ simply a "definition" for the vector $u_1$ that is related to $v_1$ through transformation by $A$. I can't quite grasp the connection between the column space and row space here, so I think it must be a definition. I know that the row space and the null space are orthogonal and have a zeroth vector in common; the same is the case for the column space and the left nullspace. But how does one link the row space and the column space?
Am I correct in assuming that $u_1=Av_1$ is simply a definition for $u_1$?