Let A be a 4x3 matrix and let b and c be two vectors in R^4. We are told that the system Ax=b has a unique solution. What can you say about the number of solutions of the system Ax=c?
A "4 by 3" matrix maps a 3 vector into a 4 vector. More specifically, it maps all of to a subspace of , of dimension 3 or less. If the dimension were less than 3 then A would have non-trivial kernel and more than one vector would be mapped to a specific vector in that subspace. Saying that Ax= b has a solution means that b is in that subspace. Saying that Ax= b has a unique solution means that a unique vector is mapped into every member of that subspace. So there are two possibilities for the number of solutions of Ax= c depending upon whether c happens to be in that subspace or not.