What is a subspace and a subset and how are they related?

[goomer]

I'm having a hard time understanding subspaces and subsets in linear algebra.

So what I'm getting is that a subset is any set of vectors in a plane R^n.

So a subspace is a set of vectors in a subspace?

http://postimage.org/image/6e2tl1c51/
http://postimage.org/image/6e2tl1c51/

^This is my thought process; correct me if I'm wrong.

 

[Poopsilon]

A subset of R^n is just some set of vectors of R^n, possibly all of R^n, possibly empty. A subspace in linear algebra, called a linear subspace, is this:

Theorem: Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if it satisfies the following three conditions:
The zero vector, 0, is in W.
If u and v are elements of W, then any linear combination of u and v is an element of W;
If u is an element of W and c is a scalar from K, then the scalar product cu is an element of W;

(taken from wikipedia)

Thus all subspaces are subsets but not all subsets are subspaces.

 

[Poopsilon]

I think the important part to take away from this is that subspaces can be regarded as vector spaces in the their own right, simply by ignoring the larger vector space they are embedded in. For instance consider R^2, this is all vectors of the form (a,b) for a,b in ℝ. Then all those vectors of the from (a,0) are a one-dimensional linear subspace of R^2, and by writing them as (a) we now are looking at this subspace as simply a vector space in its own right.