Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset? Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does?  Since it's a subset? A little unclear about this...

Less formally, but to get the idea in words : for a set to span a vector space is a sufficient condition. It means that you have enough "good" vectors in the set for it to span the vector space. If you add more vectors to your set, then you will still have your good vectors in the set, so it will still span the space. If you remove a vector from the set, however, you might remove one of the good vectors, and in that case you will no longer span the space.
Conversely, if you start with a set which does not span the space, it means that your set does not have enough good vectors in it. If you remove vectors, then certainly it will stil not have enough good vectors, so it will still not span the space. However, if you add vectors, then it is possible that the vectors you add are good vectors, and that the resulting set will span the space.
