Ok, so I understand that a vector space is basically the span of a set of vectors (i.e.) all the possible linear combination vectors of the set of vectors... I don't understand the concept behind a subspace or why it's useful. I know the conditions are: 1. 0 vector must exist in the set 2. If you add two vectors in the set together, you should get another vector in the set 3. If you multiply a vector by a scalar, you should get another vector in the set. Do conditions 2 and 3 combine? In other words, can the conditions be rewritten as 1. 0 vector must exist 2. A linear combination of some vectors gives another vector in the set ? Also, graphically, what is the subset supposed to mean? It seems like the only way for something to be a subspace of Rn, for example, would be to be the vector space Rn... Could someone give me an analogy to spark some intuition...because this seems very abstract?