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. :)
 A: The number of axioms is subject to taste and debate (for me there is just one: A vector space is an abelian group on which a field acts). You should not want to distinguish by noting that there are different criteria. Actually, there is a reason why a subspace is called a subspace: It is also a vector space and it happens to be (as a set) a subset of a given space and the addition of vectors and multiplicataion by scalars are "the same", or "inherited" from that other space. So this way there is no real difference, and one should better introduce and define the notion of subspace per "vectorspace that is contained (the way I describe above) in a vector space" instead of "subset with operations that have some magical other properties". Rather the fact that "nonempty and closed under multiplication and addition" are (necessary and) sufficient conditions for a subset to be a subspace should be seen as a simple theorem, or a criterion to see when a subset of a vector space is in fact a subspace. It gives you a simple recipe to check whether a subset of a vector space is a supspace.
A: 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 $V$.
A: Vector space contains the 10 axioms and act under those axioms. To prove some new mathematical operation or set is a vector space, you need to prove all 10 axioms hold with those mathematical operations.
Instead, you can show the mathematical set is a non empty (as it must contain at least the zero vector) subset of an existing vector space, that continues to be closed under scalar multiplication and vector addition. In doing so, there's a theorem that shows this subset is a subspace and would be itself a vector space (meaning all 10 axioms hold).
Hence, it would obey all 10 axioms of a vector space, but you only have to show a proof that it is closed under linear combination (scalar multiplication & vector addition) to hold all 10 axioms.
The zero vector existence is implied by the existence of being closed under scalar multiplication as you can just set the scalar to 0 and multiply all vectors to 0 to get the zero vector.
