Vector Space vs Subspace

Can someone explain the difference between a subspace and a vector space? I realize that a vector space has 10 axioms that define how vectors can be added and subtracted. I also realize that a subspace is closed under multiplication, addition, and contains the zero vector. My problem is that I fundamentally don't understand the difference between them.

Perhaps you guys could show me some examples of both a vector space and subspace. I'm a visual learner. Thanks. :)

• What is a subspace of a vector space? Can you guys give me some examples? – user127662 Mar 31 '14 at 6:01
• Also, what is a basis in a general vector space? – user127662 Mar 31 '14 at 6:03

Consider any $\bf{set}$ $A$. What do we mean by a subset $B$ of $A$ ? It is just a part of $A$. Now when we talk about a subspace of a vector space (or subgroup of a group) it hust means that it is a part of the vector space carrying the original algebraic structure of the vector space, i.e., suppose we take any arbitrary subset $S$ of any vector space $V$, then it is quite possible that addition of two elements in $S$ need not be in $S$ or multiplication of an element in $S$ by any scalar might not be in $S$...but when we say that $S$ is a subspace of $V$ then we mean that all the axioms that hold for $V$ also hold for $S$, under the same operations as that of $V$.