We travel deep into Melvin the Course Difficulty Dinosaur's midsection to get hold of spanning sets of vectors, bases, and dimensions. The monks have prepared the way: Our definition of linear independence of a set of vectors corresponds exactly with finding the Nullspace a matrix formed by the same vectors. Ka-ching. of The story is general but our mission with defining bases and dimensions is to fully describe the four fundamental subspaces for $\mathbf{A}\vec{x}=\vec{b}$.